Welcome to the sport. It can get addictive in a hurry. You've gotten some good advice and I would add that you want to meet and get to know some of the experienced houndsmen around you. Find a UKC hunt nearby and go watch. Meet some the members of the club and just absorb all the info you can. Most guys are real nice and are always happy to help out someone that wants to get into the sport. You can sometimes buy some good used equipment from them that they aren't using, and you may also find the dog you are looking for from one of the guys you meet. I bought a tracker when I first started hunting and a week later at a hunt I could have bought a used one for about half the price of what I paid. Don't believe everything you hear though because a lot of bull is spewn about at hunts.
As for finding a dog that hunts coons and cats for cheap - good luck. They are out there but finding one is going to be tough. Cat dogs that are finished are real pricey. If it was me (and it was about a year ago) I would start with a good coon dog. I tried doing exactly what you are talking about and came to the conclusion that I needed to focus on one or the other. Trying to do both doesn't do either justice.
Just my thoughts. Again, welcome to the sport.
Steve