$250 may not get you much in local stores but you can get some nice KICKERS from ebay. Are you planning on running one sub or a pair? What size 10" or 12"? IF you chose to run two woofers they will only fit if you use a sealed enclosure. Sealed boxes require less volume than ported but do not provide as deep of bass as ported. If you chose to use only one sub you can use a ported box. Ported boxes have a higher output per\watt and deeper bass, but the bass response is not as linear. SO TO SUM THIS UP, if you want deep ground pounding bass that can be felt & heard for blocks get one 12" sub and use a ported box. If you listen to a lot of rock and want a good linear bass response that covers a wide range(like drums) get a pair of 10"s(or 12"s if they will fit) and use a sealed enclosure.
Actually sealed boxes go lower than ported. A ported box below the tuning frequency really starts to drop off, while a sealed box will continue to extend into subsonic frequencies. You are correct, however, that the response curve is much more peaky and overall output is higher. It will give you the ground pounding bass you're talking about at the expense of sound quality and space, while sealed boxes give you tighter, punchier bass and allow your woofer to handle more power.
As far as the $250, it will get you a decent pair of kickers, but you've forgotten completely about the amp, wiring, etc. To run a 500w system (think a couple of CompVR's) you need at least 4 gauge wiring. Start going higher (L5's and up) and you're getting into 1/0 gauge with larger fuses. That stuff is expensive. Altogether you're looking at probably $100 in just wiring, connectors, etc. Additionally, you really do need to get a good, efficient class D or class BD mono amp. This is not something you want to skimp on as these amps are more efficient and draw less current from your alternator for the same output, meaning less dimming of lights, etc. A quality 500w amp is going to run your at least a couple hundred. I'd recommend rockford or kicker amps. I've found rockford to be slightly cheaper, and when you get into the power series they have built-in fan cooling. Add all this up and by the time you get it all and get it shipped you're looking at ~$500, and that is installed yourself, building your own box, etc.