Help me out with my budget rig

What kind of connections does your head unit have? There are no high-level inputs on that amp that I can find, so you will need front and rear pre-outs to use it.

The speakers....I have never heard them, but I have yet to meet an Alpine product that sucked. Have you listened to them?

The sub....I have no experience whatsoever with Rockford Fosgate, so I really am loathe to comment, but it looks kinda cheap to me.

Ok, so you're concerned about costs. Very reasonable request. Seriously though, you've got to have an aural vision in mind? What exactly are you expecting out of your system? What do you listen to? Anyone can hit a $500 budget, but unless you state what you demand out of your system, no one is going to care what they put in your car.

Standard head-units (aka, OEM) suck. All of them. I've yet to meet one I've liked aside from the fact that they have an 'OEM appearance.' You need to consider a new head-unit. What we're doing here is not simply spending your money, but laying THE foundation of a better audio system. Much care should be taken here. Specs, features, mechanical quirks, and the general concerns about how this thing will work in your ride should be well evaluated. This is the lone piece of equipment you will interact with the most, and the one you will curse the most if you don't make the best choice.

Secondly, you'll need some speakers. Do you know what kind of listener you are? It's my experience that there are two types of audio people: forward & flat. Forward is for the folks who like a bright sound, and flat for those who like the laid-back even sound. Find speakers to match.

Overall fidelity steps up next, and amplifiers are the deciding factor. When the factory head unit's power just isn't enough, or you want better resolution, you need an amplifier. Buy one with plenty of headroom. It won't hurt to over-power your speakers. As long as they aren't taxed with low frequencies they weren't meant to produce, you should be fine. In other words, if you use the appropriate crossover levels and put each speaker in the best position to exhibit its best characteristics, you'll be good to go.

I actually recommend subwoofers lastly. They can improve or ruin a system. Remember the 'Bass Cannon?' Thumbs down. You need to know what your system in a 'full range' setting can do before you opt for more bass, because excess bass is the #1 way to screw up an otherwise tight, clean, & accurate system.

Now that I've laid out the foundation for a good car stereo system, go do some more listening to components.

The lower end Alpine's have only 1 set of Pre-outs. To amp your speakers the way you were planning would require at least 2.

What kind of improvement do you want? More bass? Louder/more dynamic? More detail? You MIGHT want to consider skipping the amp, sub, or both, at least for now. The nice thing about audio is that generally you can assemble as system piece by piece, and you can still get a good improvement at each step. Just because you are going to amp your speakers eventually does not mean you can't push them with the head unit until you get a bit more cash.