Two questions in one post...
 

1. Well, me and my band are thinking of recording soon, and we can do one of two things:
a.) record with a computer microphone (a cheapy 5-8 dollar one)
b.) use one of those funky things that allow you to plug a regular mic (cheapy 10-15 dollar one with built in chord, the end being a 1/4 inch mail plug) in to your computer.
Which would sound better?

And any other tips for having the best quality of sound for a home recording would be appreciated too.

2. Without buying a PA system, what is a good, affordable speaker/amp that I can plug a microphone into (with 1/4 inch mail plugs) and have it sound nice and loud?

I hope I have my terms correct for these questions.

1 - both options are going to sound like 'crap'. but crap is a relative term. i've got hundreds of jams recorded in a room in the back of a warehouse with a built in mic from a mac laptop. it's fine for personal use, but not really good enough for a proper demo.

2 - any instrument amp will do, again quality will suffer but it will be loud.

the only term you got wrong is mail, it's supposed to be male, same as you can get cables with female ends. yep...

i'd also advise taking some serious time to set the levels on your instruments when using a single mic recording method. it's really hard to tell what's too loud in the moment though so you'll have to record first then go back and adjust, then do it again and again.

use the drums as a point of reference and adjust everything else accordingly based on what you hear from the recordings. it might sound 'wrong' while you're playing but it will make a significant difference in the recordings. the position of the amps relative to the microphone will also affect the recorded sound,

Yeah, we're probably going to test it all out by playing a bit of Blitzkreig Bop, and then seeing how it sounds, and fiddling with the amps.

Yeah, we're not really trying to make a really good quality thing, just kind of some reference material, or some stuff to give and show people.

Our method is going to be kinda like this:

Record all of the rhythm instruments (rhythm guitar, drums, bass), then record the lead guitar or the important guitar parts, and finally record the vocals.

For punk/pop punk it won't be too bad. But if we were recording something more complex, yeah, then it'd sound awful.

that should work well enough to get a basic idea out there. as lame as it is playing along to a metronome makes multitracking so much simpler.

Update: I got a new mic, a $30 Apex 850.

It's a mic with one of those XLR chord things, but I got the end of the chord that plugs into the amp replaced at the music store where I bought it with a 1/4 inch male plug.

Think I should use it to record into my computer (using a 1/4 to 1/8 inch adapter of course)?

there's no real reason you can't but every additional step in the line decreases overall quality.

are you going lofi on purpose? no budget?