Do you know any cyclists? One of the best ways to find a decent bike in that $200 price range is to purchase used. Having someone around who knows what good equipment looks like could help in that regard, and you could pick up a pretty decent ride without spending too much money. The right $200 used bike can be just as good as a new $500 bike. The wrong one can ruin your day though, which is why having an experienced friend can help. Craigslist sometimes has good offerings, as do other local used lists.
I would not suggest buying your first bike online, there's a lot of uncertainty over not riding something before owning it, and buying a one size bike unless you happen to fit that one size. Fit is important for comfort and control. You can look up how to find your proper frame size online, this is an aspect of buying a bike that I really can't stress enough: be comfy on whatever you get!
If you're not mechanically inclined, try forming a relationship with your local bike shop too.
I would pick one where the staff aren't going to thumb their nose at a newbie, and where they carry beginners stuff as well as more advanced bikes. If you're not into buying new, they probably won't have anything in the $200 range, but some stores start to stock some nicer stuff around $350, the price tag comes with a mark up, yes, but it also comes with the help that a knowledgeable staff team can give, a properly set up bike, and usually a maintenance tuneup later on. The price tag is a bit higher, but the risk of a poor bike with a short lifespan is much lower. The *right* bike shop can not only get you the right bike, but help you progress in skill and knowledge.
Finally, buying a $200 bike new can be okay too. There are some things to consider though: as before, fit is important. If you have a cycling friend, bring him/her along. They can tell you whats going to work and whats going to fall apart. None of the bikes in this price range will be performers, and some of them will be junkers. The right one though will serve you just fine for getting back into the sport, set up properly they roll just fine which is what you're after: problem is, sporting goods shops and such who sell bikes in this range often don't know how to set them up properly. I would suggest having them tuned up by professionals after purchase.
If you don't have any friends who are big on cycling, try looking up local cycling groups. A lot of them are pretty casual, and will help you out in getting started.
Lastly, it sounds like there was something wrong with the wheels/tires/tubes on your BMX. With everything on your wheels properly set up, flats shouldn't be happening on their own accord. Plus, with a patch kit, a couple of tire levers and a compact pump you'll not have to worry
Hope that helps a bit.