Will Garden Paws be ported to Nintendo Switch?
This is probably the most asked question in the past few weeks.
The devs have confirmed that Garden Paws will come to the Switch!
They are aiming for the first quarter of 2020. More details are not available at this time.

Will Garden Paws support mods in the future?
The devs have confirmed that they will be adding in Steam Workshop support after the Switch port has been released. More details are not available at this time.

How do I tame an animal?
You need to feed the wild animal its favourite food for a set amount of consecutive days.
If you miss feeding the animal for one day, the counter will go down by 1, so make sure you feed it every day until it's tamed.
After the animal has been tamed, it will follow you. You will need to lead it to its habitat. Once it goes in there, you get the option to name it and you can take it out as an item if you should wish to.
The type of feed and habitat depends on the kind of animal.

Do I need to feed my tamed animals?
Feeding tamed baby animals will grow them into an adult animal. The amount of days they need to be fed depends on the kind of animal. Contrary to taming, these days don't have to be consecutive. Failing to feed your baby animal every day does not give any penalty.
Feeding tamed adult animals has no purpose (other than roleplaying).

When I feed an animal, it doesn't register as fed.
That's a known bug that can be remedied by opening the habitat and dragging the feed onto the animal rather than feeding it manually.

Does petting an animal do anything?
No, it's just a cosmetic / roleplaying feature.

My animals have stopped producing / I can't milk them anymore!
This is a known and common bug that should easily be remedied by taking the animals out of their habitat and putting them back in.

I visited someone else's world, and now all my quests are gone!
This is a known bug - your quests aren't gone, they have actually been completed because the game wrongfully synced your quest log with the host of the world you visited.
Restarting the game should fix this issue. If it doesn't, you might need to roll back to a previous save.