There are lots of "good" bows out there. Being on a budget will limit what you can get your hands on.
There are Bear, PSE, Parker, Diamond, Martin, Browning and many others.
There are more top of the line brands like Matthews, Bowtech, Hoyt also. You could find a used bow in one of these brands also. many times, someone selling a used bow will sell it ready to rock and roll with sights, rest, arrows and the whole nine yards. This may be a better option for you.
If you can find your way to a pro shop and explain your circumstances, they may help you out.
If you can, you need to take a few shots with some bows to get a feel with the whole thing.
To me, one important thing is how a bow feels in my hand. It just has to feel right.
As you know (or maybe not) Matthews makes a top of the line bow. But I personally do not like the way they feel in my hand.
Try to find one that has "the feel"
Try to find a pro shop or archery range in you area and hang out there and try to learn by watching others. Ask questions.