The "front shock no longer bounces" is like saying the car runs fine but has no brakes. If you want to "try a hand at mountain biking," you'll want a front fork that "bounces." Also, the quality of the bike doesn't matter if it doesn't fit you. If the bike doesn't fit, you won't ride it.
If you're REALLY hard up for cash, shop at garage sales, thrift shops, or Police auctions. There are LOTS of nice bikes to be had for as low as $5. Those prices leave you lots of $$ for repairs, adjustments, etc.
The less you know about bikes, the better off you are getting a simple bike. For short-distance commuting without a lot of hills, there's lots to be said for the simple coaster-brake bike (the one you backpedal to stop). If you must have multiple gears, the internally-geared 3, 7, and 8 speed hubs are more reliable (but much more expensive). If you have long distances, steep hills, or want to try mountain biking, buy a mountain bike. Their derailleur gearing systems (look like a angled corn cob of gears on the rear wheel) take the most maintenance & adjustment, but provide lots of gears for less money.
Before deciding on the bike you want, why not go by the local bike store (NOT the department store) and try looking at & riding the various types of bikes they have. Who knows - the shop might have some used bikes or know of a customer who wants to sell one. If you can buy a good used bike (even for more money) you should be able to sell it when you're done for close to what you paid. Having the "better" bike means better reliability. That alone may justify the price. Also, the shop won't let you leave with a bike that doesn't fit.
Happy shopping!
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Nishiki road bike, Raleigh road bike, Electra Cruiser Lux 7d, Electra Townie 3i, Electra Townie 1, Whatever I find today!