Can I catch bacterial infection from my pet?

Yes. It depends on the pet. Reptiles can spread salmonella. Birds can spread psittacosis. You can get toxoplasmosis and q fever from cats. You can get leptospirosis from dogs. Also bites and scratches can cause infections.
Yes, pets have germs. Bacterial infections that a person can get from his pet include campylobacter and salmonella (both bacteria cause fever and diarrhea), cat scratch disease, etc... A variety of viral diseases, and parasites, can also be passed from pets to humans.