If you're going to get an amp and subs, make sure you buy yourself a quality head unit for the car. If you try to just hook up your stuff to a stock stereo, the sound quality will suffer. Brand-wise, I recommend Pioneer for the head unit. You can get a decent one for less than $200, and they are the best (IMO) in that department.
For amps you could go a variety of ways, but I've been using a Sony Xplod for about 12 years now and it still works well. Which brand of amp isn't as crucial as it is with head units and subwoofers, as long as you get the right amount of wattage. Any of the major brands (Sony, Kenwood, Pioneer) should be fine.
With subwoofers, it really depends on the type of sound you want from your bass. If you prefer a low, vibrating rumble then go with a larger sub (12", single or dual). If you'd rather have a hard-hitting, low resonance bass pound, smaller subs are the way to go (dual 8" or 10"). For manufacturers, I suggest you stick with Kenwood or Alpine. Audiobahn/Dual is good if you can find them, but they're generally rare these days.
Now, onto the specifics. When shopping for the pieces, you need to make damn sure your wattage and RMS on the amp and subs match up. If you get a high wattage amp with a low wattage sub, you'll blow your subs. Vice versa, if your amp is underpowered in relation to the subs, you'll blow your amp's power supply (learned this lesson the hard way, I did). A good number for an amp is 1000 watt RMS, and the subs should reach a similar number cumulatively (if you have two subs, get them 500 watts each, if you just have one get a higher wattage). Also, make sure you look closely at the numbers, because a product might say it is 1000 watts, but that refers to the maximum potential and doesn't reflect its average usage. You never want either of your products to be operating at max power for extended periods of time.
In addition, when shopping at Best Buy or other places, don't be afraid to jump on those "open box" specials. That's how I grabbed my Dual 12" subs fully assembled for $90.
$500 can get you all you need, but $600 would be a much better target to make sure you get quality stuff. Remember, you're not just buying sub, amp, and head unit. You have to buy all the wiring, kits and harnesses to hook everything up, which can easily cost you another $50-100. If you decide to have someone else hook it up instead of doing it yourself, you also have to factor in fees and labor. I know you want to stay under $500 and you absolutely CAN, but you'd be going with lower end items that won't have the longevity and pure sound quality that the better brands have. I can promise you that you will get your money's worth out of the suggestions I made, but they'll probably put you a bit over your $500 budget.
On a side note, if you have the cash and you really want the best quality sound in your vehicle, replace your stock speakers with Alpines. Alpine is the absolute best brand for speaker quality, and aftermarket speakers make a great combination with an amp and subs.
So, for the tl;dr version:
Subs - Kenwood or Alpine, 8-10" for a harder hitting bass, 12"+ for a deep, resonating rumble. Don't spend any more than $150-200 for a set of two. ~1000 Watts RMS for power capability (total).
Amp - Sony, Kenwood, Pioneer, 1000 Watts RMS for power, don't spend more than $150 here.
Head Unit (optional, but HIGHLY recommended) - Pioneer, period. Don't spend more than $150 here. Make sure it has FRONT side aux input or ipod integration (back side aux is a royal pain in the ass). Large video screens are obnoxious and a waste of your money, don't get them.
