I've only been doing surveillance for a few months. Mostly workers' comp.
I use a Sony DCRSR68 handycam. It does most of what I need. Easy to handle, decent battery time, more memory than I could ever use, it has a decent zoom and the image quality is good.
It's fairly useless in dark situations though. I will definitely replace it when I have the spare cash. Look at the 'Lux' rating on the specs when you're considering a camera, the lower that number the better it works in the dark.
I think most of the consumer level handycams will get you started.
Features that you absolutely need to look for are time & date stamp on playback and an AV-out jack, to run it into a DVR and get you product onto a disc. I wasted my money on a JVC Everio that dosen't put the time/date on the footage :evil: , the best footage in the world is useless without it.
Apart from that you'd need something for close range, like follows into the supermarket etc. I made do with my handycam in a box for a while :oops: but I've got a Lawmate PV900 on the way :mrgreen: I tried a $200 mini-dvr with buttonhole camera from Jaycar but it was unreliable crap so I took it back. Take the advice that other people have posted here and steer clear of budget bagcam/bodycam units.