Anyone have real experience with cats (maybe even dogs) as pets?

Most cats throw up. It's normal. Carefully examine what is in the sick.
They swallow their own hair when licking and grooming themselves. It's not digestible, so they chew a bit of grass to help them vomit up the hair.
It's possible to reduce this to almost nil by grooming the cat every day. They absolutely love it.
(But if you have a long-haired type and if that hair has become very matted, it would be far kinder to get the cat sedated and shaved at the vets. Then start the grooming from the first day home and keep him/her warm while the hair regrows.

Don't overfeed. A cat's stomach is only the size of a golf ball and if they eat too much they are very likely to sick it up. Better to feed two or three fresh meals per day, and remove whatever the cat doesn't eat. (I give leftovers to the birds, dusted in DCP (dicalcium phosphate).)

If you don't see hair in the vomit, then the cat may have a food allergy. Most adult cats are allergic to milk, and even the commercial "cat milks" may still cause vomiting. Kittens need special milk - never cow's or goat's.
If milk is not the problem, then you may need to do an elimination diet to find out what the cat is reacting to. Lamb is often one of the most easily tolerated foods.
A home-made food is always far healthier than any of the manufactured brands - but it's costly and time consuming. I make a month's worth at a time and freeze them in golf-ball sized portions. One can look up the ideal natural cat diet online - or you can message me and I'll send you the recipe I found which worked very well. The most important thing to add is taurine; cats need it for good eyesight.