I want to branch out from PLA. Should I try ABS or TPU?

I would try PETG. It doesn't need an enclosure.
- For smaller prints, agree. For larger prints a cardboard box will do just fine, but it needs a little something in my experience. Not too much mind you.
- How is PETG different (better?) than PLA?
  
  It can handle higher temperatures, and UV rays and is not as brittle as PLA. It has a little flex.
  
  It is better in every way except easy printability and water absorptionon the spool (wet petg Is much harder to print but it can get wet after printing no problem)
  Stronger, UV resistant, more watertight, much more heat resistant, more resistant to creep at room temp, less brittle, can print clear, and doesn't have a bad warping problem or need and enclosure like printing ABS.
  It is essentially just PLA but better and a bit harder to print. I completely switched over to PLA because I found good settings for FormFutura recycled PET (more stringy than petg)

ABS needs an enclosure for anything above around 10 layers. Even a room closed with it warm and no one inside is not enough to save an ABS print. Just the air from the moving tool head and the bed are enough to disturb a print and cause layer separation. An IKEA Lack table and a garbage bag over it is enough of an enclosure to count and get most prints alright. It stinks though.
TPU will have holes and look terrible unless you print out of a filament drier. You can dry the stuff a lot and print for around 45 minutes with it in open air before it will absorb enough moisture to start expanding steam in the melt zone and blowing holes in your print layers.
- That enclosure might work for smaller ASA prints, but I needed a lot more insulation than a garbage bag to pull off larger prints.
  Maybe I got lucky with TPU, but I didn't run into any significant issues with humidity when I printed treads for wagon wheels over the course of two or three days.
  
  Makes a huge difference with my junky TPU if it is dry or not.
  I also have a massive stacked Lack (sp?) table with double legs. I put that stack with a MK3S+ into a 55 gallon trash bag and can print the entire build volume. The thing is, it has to be totally sealed off well and no traffic in the room that might disturb the thing in the slightest. I won't even open a door to the room. I also let the bed heat for longer before the print starts. Lastly, I must design for ABS specifically and am very conscious of layer thickness transitions. I look at all filaments and designs as an optimization exercise for materials and process kinda thing. I design everything I print. So my advice is an abstraction of what is possible under similar constraints. Most files people share are not very well designed for 3d printing, or for material specific requirements/optimizations. I don't recommend printing other people's stuff unless you are forced to for some reason.
  The primary issue with ABS is how heat is soaked into top layer/bottom layer transitions near any side walls. In most cases, just make a tapered transition over a long area and remove any top layers in places like interior surfaces. Designing tops sections that are rounded or hollow is another key, along with no supports based designs. Using exposed 3d cubic infill is how I get around a lot of the top layer heat issues.
  If anyone is actually wanting to print ABS a lot, obviously just get a Voron.
- Abs needs an enclose vented to the outside, or really good ventilation. The 'S' in ABS stands for styrene, a fairly potent carcinogen and irritant. Do NOT print it in your living area as it off gasses while printing.
  All other plastics you can print on the A1 will be safer.

IMO having the ability to do TPU is way more versatile than going to another rigid structural filament.
ABS/ASA is just "pla but more impact resistant".
TPU is "haha funny squishy wait this turns into a living hinge?" and opens up a TON of print opportunities.
I had a lot more fun trying out TPU (both high and low durometers) than switching to any other kind of filament. Whatever you print basically becomes shockproof l, is squishy/bendy, and you can chuck it across a room full-force with no problems. Super fun.
However, TPU is happiest with a direct drive extruder. High durometer (95a) TPU's are fine, but not optimal, in bowden extruders, while low durometer (Ninjaflex) straight up won't print right thru a bowden. So keep your type of printer in mind when shopping for spools of test filament.

It really depends on what you're looking for. Are you just looking to learn how to print new materials, or do you have specific requirements for a project?
If it's the former, I'd say the easiest thing to try is PETG. It prints pretty reasonably on most printers though has stringing issues. It has different mechanical properties that make it suitable for other applications (for example, better temperature resistance and impact strength). It'll be much less frustrating than trying to dial in ABS for the first time.
ABS and TPU are both a pretty large step up in difficulty, but are quite good for functional parts. If you insist on learning one of these, pick whichever one fits with your projects better. For ABS you'll want an enclosure and a well ventilated room (IMO I wouldn't be in the same room as the printer) as it emits harmful chemicals during printing.
- I designed and printed a fender for my kid's bike. It would be totally usable as a fender but he's a savage so he broke the first one within 30 seconds of me installing it. Then I changed the design to add more support and that one lasted a full hour before he broke it off. So I'd like to print the same design again in a new material. I think either of these two would work, as they would put up with more abuse in different ways. one would be much harder and the other more forgiving.
  I don't love the idea of the toxic fumes. I don't want to get an enclosure and the printer is in my home office off of my living room. I could crack a window and let it run overnight but that sounds inconvenient.
  
  TPU will be nearly impossible for your kiddo to destroy. ASA/ABS are rugged, but if your print has thin surfaces it's less strong in my experience than PETG.
  I didn't find TPU hard to print personally, just go slow and turn retraction way down or completely off. It will string pretty good, but most slicers have a setting to avoid crossing perimeters that will keep it in check. The only thing I would be wary of is ending up with a floppy print, so make sure the part has some structure.

I'd say TPU
It's a lot less gnarly than ABS and depending on the hardness you get can it can be anything from a pain to a cakewalk to print.
Plus IMO it's got more uses due to it's durability.

I personally would start with TPU(after PETG) - the different shore grades provide a large usability for a lot of things and it print comparably well once you find the right calibration. Especially with a A1 as you mentioned you have one.
To be honest I would skip ABS totally in an A1 unless you have an external enclosure with a good filtering and exhaust solution. Be aware of the noxious fumes ABS will produce that have a potential to intoxicate you and are suspected to cause cancer depending on the additives. (Among others ABS produces hydrogen cyanide when printed - which is often better known under it's former German brand name: Zyklon B....)
ASA nowadays provides a far less problematic (but not unproblematic) solution and while it's a little bit more complicated to print it's still manageable depending on the filament manufacturer. But you will need a temperature stabilised enclosure for both anyway, while ASA is a bit more sensitivitie,it doesn't really matter that much for me.
Within ASA I personally found a far larger bandwidth of printability between the manufacturers. The major manufacturers for PLA often suck - especially Bambu Lab ASA is hideous to print. If you are in Europe I cannot recommend the Black Forest Filament ASA enough, their stuff is not comparable to any other ASA I printed. Alternatively material4print. If you need a filament that is available worldwide Filamentum Apollo X is a solid choice, so is Polymaker,but both to a lesser extend.
In theory PMMA, PCTG and CPE are also worth a consideration, but besides CPE all of them are far more difficult to print.
BTW: All variants need to be printed very dry, ideally out of a warm dry box.

TPU is a fun one. Touch, extremely flexible, but not stretchy.
Don't try to feed it with a bowden tube; you need a direct drive extruder. Other than that, it's incredibly easy to work with.
- I've printed plenty of different mid density tpu's including random no name ones with generic profiles, using the Prusa Mini with its Bowden tube. It's not always a no no.

Petg or Tpu
Fear ABS and ASA like your worst nightmare
- Fear ABS and ASA like your worst nightmare
  ABS and ASA are awesome, but if you don’t have an enclosure the warping and odor can make them a pain to work with (the odor is powerful, irritating, and potentially dangerous).
  But the end result is strong and can be smoothed with acetone fumes. I consider ABS and ASA my “okay, this one is serious” filaments. Parts that need to be able to withstand stronger forces and higher heat, or stuff that I want to look really good with some very easy post-processing.

TPU is what I'd suggest. You don't need a enclosure and it should work for what you want. It's a finicky material though so be prepared to spend some time getting to know it, figuring out it's temps and speeds is a must for good prints.
- Depends on the hardness of the tpu. You don't have to dive into full flexibility

TPU.
ABS is a beautiful filament, but it’s somewhat similar to PLA in its properties.
TPU I fun to play with and its flexibility offers different properties than PLA

If you have an enclosure, I would say try out ASA. Similar to ABS, I've found it makes some really nice parts that are tough and UV resistant. Great for things you want to leave outdoors. Otherwise TPU is useful stuff too, and no heated chamber required. I was using it this weekend to print up seals and grommets for a trailer I'm rewiring. It's nice being able to print up some parts I need rather than making a drive to the hardware store and hoping they'll have something I can use.

I would set my goals to mastering PETG next. It's easy to get and inexpensive in price. It offers more toughness, (do not confuse toughness with hardness), plus it's more heat and UV resistant also, (though PLA has surprised me with several outdoor repair parts I have made) . But it can suffer from creep in assemblies. It can make durable car accessories like phone holders or a cup holder cheap and easy to design.
ABS/ASA really needs a heated enclosure to print reliably day in and day out. And the fumes they give off during the print process are dangerous to your health. And you must have proper ventilation to use them safely. I keep as spool around, but I very seldom find the need for ABS anymore.
TPU, while a fun material to play with, is pretty much a one trick pony. The vast majority of things you will print will seldom require TPU. I try to very hard not to print just one item in TPU because it's not worth the effort. I try to line up several projects that need TPU to make it worth my while because there is hours of drying time involved before printing.
If you want branch with different materials, you will need to add a filament dryer. You can get a purpose designed unit that can quite spendy or you can get a cheap food dehydrator and get the same results in the same amount of time.

TPU really needs direct drive extruder to work well. ABS needs an enclosure, but you could just make one from an 8x4 sheet of foam insulation and tape for cheap.
- I don't know what a direct drive extruder is but I have a Bambu A1
  
  A direct drive extruder is one where the motor that runs the filament sits right above the hotend. Since TPU is squishy, the long tube from a Boden setup gives it too much room to squish.
  
  That's a bowden extruder, you'll have problems with TPU.
  Try PETG, just dial up the retractions a bunch and slow it down.

Adding to what others have said: You can print TPU with Bambulab A1. You just need to feed it directly to the extruder and not through AMS. The generic TPU settings in BambuStudio/Orca should be fine, but make sure to run some tests and slow it down if necessary. If you need supports and have trouble removing them try printing support not in tree mode, but the grid one. They should peel off easily. You should also give PCTG a go if it's readily available where you live. It prints like PETG, but it's stronger and more heat resistant (which you'll need if you want to print a reusable spool)
- You just need to feed it directly to the extruder and not through AMS.
  Though Bambu Lab’s “TPU for AMS” works great (as long as it’s dry).

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