Yes. You can see what abilities a certain beast will have by casting beast lore on it. The only abilities you will get at a pet trainer are growl, great stamina, natural armor, and all of the magic resistances. So if you tame a level 10 cat, for instance, it will have low ranks of claw and bite (rank 1, probably). If you keep that same pet up to 70, it will still have those same low ranks of those abilities. You need to tame higher level cats (or other beasts) to learn the newer ranks of those abilities, and then teach them to your main pet.
For everything you could possibly want to know about pets, go
here. Oh yeah, and the pink bird he's talking about is in Mulgore.
Here is a picture someone put up of the tamed (I think) version of the bird.
Edited, Jul 29th 2007 12:55am by mikeyvach