It's really hard for me to pick only a few species I like the most as furries, as I like a lot of animals, and any animals can make cool furry characters. Plus the overall design and appearance of a character can tend to matter a lot to me and not just the species.
Though I suppose I do have a strong preference for cats, or felines in general, and some big cats. To name a few others I generally like, I suppose canines, mustelids, hyenas, raccoons, skunks, rabbits/hares, pigs, cows, horses, rats, foxes, goats, dragons (especially Eastern), bleh, the list goes on. Again though some of these can really depend on the art style; for example I am not as much of a fan of cow or horse characters that have more "humanoid" shaped heads (particularly with female characters). Then again I am not a big fan of that for furry art style in general, but I tend to see it more often with these species. Same with rabbits; I like rabbits characters that are a bit more realistic in terms of facial anatomy.
In terms of more fandom-specific, made up species . . . to be honest there aren't any I really have much of a strong interest in. I guess maybe sergals, and manokits. Airplane dragons are also cool. I don't really know a whole lot of the made up species tho.