Depends on what you're doing. If operating machinery like a lathe or a mill or anything that moves and where you don't want to have body parts in the mix, no, you absolutely shouldn't wear gloves. They catch much easier than naked hands and take it from me, it's no fun getting caught. 🫠
Some protective cream is better when you can't use gloves, but I personally don't find it helps all that much. YMMV.
Otherwise yes, gloves are good. Much of the oil and shit isn't good for your skin and who knows how much gets absorbed that way.
Usually the right side has clean hands with the caption "so that yours can look like this", but OP is asexual so doesn't care about that I guess? So there's no other person involved
It's not perfect, but I use baby wipes to clean off the worst before I head in. Don't have running water in the shop so to avoid putting black fingerprints everywhere, it's good enough. Also much cheaper than specialized wipes. After that, ordinary soap usually does the trick.
Dish washing liquid is better than ordinary soap I find, but rarely necessary. A specialized product with something grainy in it is better when you're really dirty and it's on there good.