Can I treat upper respiratory infections at home?

I currently have 3 cats who all show signs of being sick.

Symptoms include:
- Lack of appetite/thirst
- "Calm" or overly mellow behavior
- Coughing (or attempting to spit something up, but nothing comes out)

This recently just happened ever since I introduced a new kitten to our household.
The kitten was in cages along side other cats who were within their own. My cats were healthy and NEVER had a health related issues until I got this kitten so I think it might have started because of it. The kitten has received de-worming treatments so anything worm related is probably out of the possible diagnosis.
I love all my cats and don't want them to be miserable or even die.
But since I have more than one, I won't be able to afford all the vet fees to get them checked out and prescribed meds. There aren't any low-cost/low-income or free vets in my area that will perform an exam.
I looked up antibiotics and other cold related meds that I think I should end up buying for them. 
I ultimately am wondering if anyone has had good luck with treating their cats at home. And if there are any suggestions for in-store meds.

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