Walls and Ceilings - Drywall cutout tool

Can anyone recommend a good tool to use
to make the cutouts for a basement I'm finishing?

Rotary? Router? Combinations?
I'm pretty cheap. :o

The original and classic is Rotozip. Bosch makes those. The basic model costs about $60 bucks. Some other company make a similar product. If you have a cummins tool store or a harbor frieght near you, you can pick up "factory reconditioned" tools for cheaper.

I have the Porter Cable "zip" type saw and have used a jig saw, but for occasional openings a small hand drywall saw is less hassle and doesn't put drywall dust in the air.

As a weekend warrior, I've recently had to work out the best tools and techniques. The key is that tradespeople will become skilled with say a roto-zip, and then swear by it as the best and only tool to use. However, I've found that it takes some practise to be fast and accurate with it... and if you have no more than a few dozen cutouts to do, you end up spending more time practising, or fixing the mistakes which are pretty easy to make with a powerful tools rotating at thousands of rpm!
Here is what I've found works for me:
Square holes: use a sharp blade and ruler.
Round holes: Get a good quality hole scorer. This is like the old compass you used at school. It has a sharp point for the center and an adjustable cutting wheel which just cuts through the paper. Use it on both side of the sheet, and then finish the cut with a blade. I tried the type of drywall hole cutter that attaches to a drill, but I didn't like it. The manual one is more accurate and makes a cleaner cut.
Doorways: the rotozip is nice and fast, but does create a lot of dust. A drywall saw is a little more effort, but creates less mess.
With a roto-zip you can sometimes avoid measuring and pre-cutting the hole: mark the center position of a cutout on the sheet. Put up the sheet by a couple of screws then finish the hole in situ using the rotozip. - start from the center move outwards till you hit the edge of the fixture. Jump over the edge to the outside edge and follow it around, using the edge as a guide. I've found this works with metal fixtures like recessed lights, but don't try it with plastic ones. The advantage of this is that you get the hole in exactly the right place. However, I'm not convinced it's much faster or easier than pre-cutting.
Measure twice - cut once!

Also, in a pinch I often use a normal high-speed 3/8" drill with about an 1/8" - 3/16" bit. Like the Rotozip, it does take practice, and you'll find which direction to do circles (I can never remember, I just do it). One way is easier than the other.

For most of the holes in my 5/8" drywall I used a jigsaw (after first drilling a hole for the blade). It made nice clean cuts.