Looking for something to tool around the park/levee with kids. Looking for a basic mountain(ish) bike. Pure road bike is not really an option given NOLA potholes, etc.
From what you describe your options are a hardtail mountain bike (with road friendly tires), hybrid (mountainish), or comfort/cruiser. Go to a couple real bike shops for test rides and avoid department store (disposable) bikes if you plan to actually ride the bike. Craigslist can be hit or miss if you aren't familiar with maintenance and sizing.
Big brands (Specialized, Trek, Cannondale, Giant, etc) won't have a huge performance difference at lower price points. Consider an aluminum frame instead of steel/cro-moly for weight savings and a slightly better component group if your budget allows. Avoid rear suspension and disc brakes since they aren't needed and are worthless on entry level bikes. Check MTBR (hardtail xc category) and Road Bike Review (hybrid and cruiser categories) for user submitted reviews of all the popular models.