I second all of this. I started using AntennaPod a couple months ago and like it. It handles "chapters" in podcasts better, too. Searching is fiddly though.
You have various options to export the data. One of those exports subscriptions, listened episodes and queue to another device. I have used it several times and it works well.
It's great and the main reason I first started using it. There is a database export and import option which transfers nearly everything. It transfers subscriptions, settings, history, etc. I think the only thing that doesn't transfer is downloaded episodes.
For Android: It depends on your needs. For better user experience I'd say PocketCasts. For best customization Podcast Republic, Podcast Addict. AntennaPod was for a while the only open source project, PocketCasts is now open source too. If that matters to you, go with those. I like AntennaPod, specifically the sleep timer. However it has some downsides: discovery / search sucks and it's hard to access the queue quickly if your phone uses gesture navigation.
@binarybomb@fne8w2ah A long time ago I used Pocketcasts which is very good but not open source.
FOSS options are AntennaPod or Podverse depending on your needs.
AntennaPod is very popular but Podverse is newer and rapidly developing and it includes a great web player. I'm using Podverse currently, which works with or without an account.
You could also use any RSS reader, and just use Podverse to search and find podcast feeds to subscribe to.