I suppose it would depend on how the races identify, and on how human they look. If the half-animal races consider themselves to be above the standard animals, then they'd likely share humans' viewpoint on the matter, and not object to it. For example, a catgirl might be fine with someone having a pet cat, and might even have one of her own, while a bull man might enjoy his steaks.
If the half-animal races identify more closely with standard animals, then things could get messy. In this regards, it could go anywhere from how humans treat gorillas & chimpanzees, to viewing themselves as the same race as the standard animals, or anything in between. This is especially true if it's possible for a standard animal to become a member of a half-animal species, such as the nekomata from Japanese mythology. That bull man might think of eating a steak as cannibalism, but have no qualms with pork or chicken.
If the half-animal races look and act mostly human but with animal features, then people would have an easy time seeing them as basically human, just with more noticeable racial features. Conversely, if they look and act more animalistic, people would either see them as having a different culture, or as being closer to actual animals (it could go either way). Either way, there'll likely be racist individuals who treat them like standard animals even if the only difference is that they have animal ears, and there'll be better individuals who treat them the same as humans even if they're basically bipedal cats & dogs. For a small example, consider beastmen from fantasy worlds and kemonomimi from Japanese culture: beastmen are more often treated as having their own distinct society, while kemonomimi are usually just integrated into standard human culture wherever the work is set (but generally not considered to be the same species as humans, because that would imply bestiality). For another example, Goofy and Pluto are both dogs, but one is Mickey's friend and the other is his pet; the difference is in how anthropomorphic they are.
These are only a few possibilities; depending on how you want things to play out, the possibilities are effectively infinite.
As an aside, there are some other issues you might want to look into:
- I'd be interested in how animal rights activists would act in your world, and how much support they'd get, seeing how similar some of the sentient species are to standard animals.
- Considering how racist (or conversely, how unconcerned with race) people can be when the biggest differences are skin colour, facial features, and/or language and pronunciation, I'd be curious how they'd act with much more noticeable features. Would bigots try to treat catgirls & dogmen as pets, for example?
- What would the half-animal races' stances be on eating meat, and how would global culture as a whole feel about it? Would people be more uncomfortable eating meat in mixed company, for example? Would half-animal individuals be fine with eating meat from animals similar to their animal half, or would that be too close for comfort? Would they be fine with eating meat from animals different than their animal half? Would, say, a bull man see eating beef as cannibalism, or would he be A-Ok with it?
- How would political correctness interact with individuals that could be actually be seen as animals by some people?
I would suggest taking a look at TV Tropes; they go into detail on how many plot devices and concepts are used, along with providing examples of how other works implemented them. Since I don't have enough reputation to post more than 2 links, I'm just going to provide a couple here, then just give you their names for other tropes. [If you're unfamiliar with the site, then to use the names, just paste them after the "/Main/" part of the links.]
Some links you may find useful, for looks at these things and how other works have handled them: