I guess it's your definition of "low budget" If you're really looking for "simple stuff", everything jhelmuth said....
I'd look for a Sony MiniDV camcorder if you're trying to stay with a reasonably priced option. You can get mechanical or electronic housings for it and you can easily capture video with most reasonably powered computers and inexpensive NLE's(non-linear editors). More u/w video is shot with Sony products than any other brand.
For under $1000 or less, Equinox or Ikelite are reasonably priced mechanical housing options. I'm not a fan of the basic seal and go "tube" housings any more unless money is really a concern as they're limited in their functionality. I wouldn't have said that till my last trip, when my dive buddy saw, stopped and filmed some tubeworms opening and closing in the current, without the ability to change to u/w macro focus, the shot wouldn't have been possible. If you can afford it, a good electronic housing from USVH or Ocean Images is an option for around $2000. Nice thing with electronic housings are that all the controls are in one place, so any camcorder you upgrade to will also work as long as it fits, and they're a lot simpler to setup and go. Most mechanicals have to be retrofitted if you change cameras.
There are sometimes good deals on eBay, but there's also a lot of old junk. Also Backscatter occasionally sells a good used housing.
And if you're going to film mostly in the daylight, above 60' or so, you don't need to get lights immediately, just a red color-correction filter for the housing.
Most prosumer camcorders have a still-shot option, on most of them, the images are even saved to some form of smart media card. However most have a CCD chip that's under 2 megapixels. Some even as low as 680K pixels. So the image quality will be pretty low, probably acceptable for a snapshot but not much more.
You can also capture a frame from a digicam during the capture or editing process with most NLE's. I use Pinnacle Studio9 and it has that function built in.