Mini dv camcorders are the best way to go these days for both $ and quality. Both Sony and Canon have a couple for around $349 to $400. You really don't need to spend more than that.
They use a digital tape that can hold an hours worth of film. So if you start stop with the whistle you can pretty much get a whole game, but it is better to have two tapes. The batteries that come with them last around an hour, so you will want to get a bigger battery. The batteries cost about $60 to $70 for a 2 and a half hour battery.
With the Mini dv you can plug it into a t.v. to view your tape, into a vcr to copy it to vhs, or the best thing is to hook it up to your computer via firewire cable and copy it to hard drive. From there you can edit and burn to DVD.
I work at Best Buy in the Digital Camera/Camcorder department, so if you have any other questions just post them up.