1) Open character screen, "C", and click on the pet tab, which only works if your pet is out, hover over the green smiley face in the upper left corner of the pet screen and it will tell you what your pet will eat. The descriptions are generic, so meat= any meat, mold= any mold, fish= any fish (fish are not meat) etc..
2) Pets in this game are not like pets in FFXI, in FFXI there was a big differenc in pets depending on what you are fighting because of the nature of the creature you had tamed and the creature you're fighting. In this game there is no such set of weaknesses as all damage from your pet is physical and only armor (and a few other things) determines how much damage your pet will do. Different pets have different base stats, based on pet family not on individual animals (meaning all cats, all boars, all crawlers, all bears, all wolves etc.. will have the same stats given the same set of circumstances (lvl, loyalty level, etc..))
For more information on pets go to petopia.brashendeavors.net/ and read up, theres a lot of information there, everything you would need to know to figure out which pet you would like to use.
One other thing I might add; please do not ask "What is the best pet," because you will probably only get sarcastic comments that won't help out as well as look like a noob. Also, the answer to that question when taken seriously is not easily answered, but the BEST answer for what the best pet is the pet that you like the most, aesthetically and performance wise, go with what you like, if it works for you use it. Once you get a feel for how the pets work you can ask which pet does more damage, or which is a better tank or whatnot, but until you have some experience in the pet area don't worry too much about all the details.