I *think* we are in the mid or low-end consumer grade camcorder area.
Canon HF M series. (NTSC standard 30 fps)
Case (I like Pelican - but a soft case might be OK)
Memory card (several 16 gig cards might be appropriate)
(optional) high capacity rechargeable battery from Canon or plan to use the AC adapter and power cord (remember, safety, first).
Two worklights on stands from the hardware store (and power cables)
Software: Windows: Sony Vegas - Macintosh: if less than 3 years old, bundled iMovie (Final Cut is preferred, but not required). In either case, the computer needs a minimum of 4 gig RAM, and LOTS of available hard disc drive space. An external drive just for the video project files is preferred.
Yes, the HF M is an AVCHD camcorder, but your budget does not allow for low compression HDV (Canon HV40, miniDV tape). You may be able to get improved performance for slow motion playback by adding light and manually opening the aperture/iris which will increase the shutter speed. Just increasing the shutter speed without adding light or changing the aperture will result in a "strobe" effect that can be challenging to watch. When the video editor slows things down, right around 15 frames per second is about the point where you start to notice a slight pause between frames.
Again - these are my opinions. I have captured indoor basketball games and used the video editor for slow-motion playback. The coaches, parents and athletes found the video acceptable (but I was using HDV, not AVCHD).
Use the white balance. At the very least, use the available camcorder presets.
Consider downloading and reading through the camcorder's manual available from the manufacturer's web site BEFORE you buy. Make sure what you want and what the camcorder can do line up. If you have a question, post here again.
Setting the expectation: Remember, you are working with a consumer grade camcorder. Slow motion "with great clarity" at the stated budget can be a challenge and we cannot afford a real high speed camera.