Front brake is a mechanical disc, rear is a calliper. The front brake locks up when used so will be replacing both, am used to hydraulics as I used to have them on my Lambretta when I had it. As I said earlier, only cost me 70 quid, I'll spend a bit of time and money getting it to where I want, the shocks on the front probably cost more than I paid for it, the seller didn't really have a clue.What brake is on the front? Unless that's a hydraulic disc as well I wouldn't bother putting one on the back.
Hydraulic disc brakes are awesome for stopping power but the pain in the arse factor of bleeding them means I wouldn't bother on anything other than a mountain bike. Even then unless you're planning on going near steep hills it's probably overkill.
A set of V brakes set up and bedded in correctly will give you close to the same braking performance while costing less money, time and sanity. I replaced mine this year for about 50 quid, front and back including new cables, from ChainReactionCycles and I reckon the braking power is about 85-90% of my mountain bike.
If you do go hydraulic disc get one that uses mineral oil rather than the various DOT fluids, it's much easier to handle and dispose of in my experience.
This kind of thing but made by Chilli Works:-