Okay, a few very general thoughts about what constitutes a PA system.
First off, don't underestimate what you need in terms of a mixer. It's probably the heart of your system. You'll need an input for every voice or instrument you want to put in the mix--and don't forget that some things like keyboards may want to use two (i.e. stereo) inputs. Also, be careful of the "mix" (pun intended) between microphone and line inputs. Some may be able to handle both but others may be line level stuff only. (FYI, when I say "line level" I'm thinking of things like your keyboard, perhaps the CD player you use for pre show music, that sort of thing.
However, besides the absolute basics, you also have to think about things like monitors and effects.
For monitors, you will want to use things called "Aux sends". These are the extra knobs on each channel strip and they allow you to send a feed of each channel to a separate output (or outputs) for the monitors. By controlling the level of the send on each channel you create exactly the mix you need in the monitors--maybe heavy on vocals and keyboard but light on rhythm guitar for example. If you want two different monitor feeds, you use two different auxes--and so on. Auxes are divided into "pre fade" and "post fade" which, as the names imply, are either before or after the main fader (which is used to control what's going to the main mix). Monitors are usually pre fade so changes to the mix don't affect what you hear in the monitors.
You may also eventually want to add effects--maybe a bit of reverb on the vocals for example. These can also use an Aux (usually post fade) but also require a spare channel to bring the effect back into the mix. (There's another way to do effects using what's called an insert--but this limits each effects box to a single channel. However, we can discuss these subtleties later.)
Anyway, if you think you need 8 channels now, I bet that within a few months you'll wish you had 16. Similarly, be aware of things like the number of auxes, the mix of line and mic level and all that tosh. Even if you don't understand it all now, it WILL become important as you start to use your gear.
After the mixer, you need to feed your mixes through an amp to some speakers. As you've already identified, some mixers have built in amps, while others (the majority) just give a line level signal you can feed into a separate amp. There are advantages and disadvantages to the question of whether you want a mixer with a built in amp. My personal preference is separate units just for the flexibility when something goes wrong--but, as I say, that's a personal preference. Just to add to the confusion, you can also get speakers with built in amps--you just feed an audio cable from mixer to speaker, plug the speaker into the mains, and you're good to go. Again, though, there are advantages and disadvantages.
The same applies to any monitors you want to use by the way--they also need an amp and a speaker and the amps can be in a mixer, separate or in the speaker.
Finally, as others have said, don't discount the number of mics, stands (mic and speaker) and cables you'll need to make things work. Then cases to protect everthing when you throw it into the van. It all adds up.
How much? I'm afraid that's a classic "how long is a piece of string" question. However, as a general principle, I'd suggest you plan on decent stuff second hand rather than cheap stuff new.
