The other responses are good - so pick one of them as "best" I just want to put them into perspective.
You want to spend less than $400. Take a look at what is used at the college or pro levels...
1) The "Pedestal studio cams" start at about $60k each. Then another $20k for a lens. They do not do audio - video only. Many times you will see a mic taped to the lens to pick up crowd noise. The video is transmitted over cable to the control room - typically the container part of a semi-tractor-trailer that has been made into a video control studio. There can be up to 6 of these pedestal cams at work.
2) The on-field shoulder mounted cams are in the $6k-$20k range. The wireless transmitters are about $8k each. The can do audio, but that is not usually used anywhere. There can be up to 6 of these cams on the sidelines.
3) the over-head sky-cam is around $30k. The cables and controls are another $30k. Usually one of these.
4) More audio on the sidelines is captured with a parabolic mic. There can be 2-4 of these. Big plexiglass, shoulder-supported, handheld "dishes" with a mic designed to focus on a far-away audio source - like the quarterback calling plays. These run about $30k each.
5) Place another 12 or so mics around the stadium for ambient noise. These all connect to an audio mixing board to get mixed before they are fed to the broadcast signal.
There is a whole lot more going on, but I just want to put things into perspective.
For $400, you need to get a tripod, maybe a high capacity (optional) batter from the camcorder manufacturer and the camcorder. And BEFORE the big day, PRACTICE PRACTICE PRACTICE so you understand what the camcorder can do under the conditions of a game.
Think about that... the players PRACTICE PRACTICE PRACTICE before the game - so do you.
Canon, Sony and Panasonic make camcorders in the $250 price range - their entry level. Then you need to edit... and we don't know what computer you have or what your editor budget is... which could mean more $ to spend.