I've only gotten into actually maintaining and using straights relatively recently, but I've done some research and experimentation. Feel free to take what I say with a grain (or handful!) of salt.
For starters, you'll probably want some manner of large, flat sharpening stones. In my case, I've got a 1000/4000 grit combo waterstone and an 8000 grit water stone.
Hitman's right about the pressure. I'd start on the 4000 grit for a dull razor, using only as much pressure as necessary to keep the razor flat on the stone. Depending on the state of the edge, use an appropriate number of strokes edge-forward. I usually finish each grit with a couple edge trailing strokes... apparently it helps prevent a wire edge on straights... or so I've read!
Rinse your razor off between grits too... and when drying the blade, treat the edge very gingerly. A feather-light stropping on a soft towel for example.
Anyway, after 4000 grit, I repeat the process on an 8000 grit. Winding up this step, I try to use lighter and lighter pressure, till I'm practically just guiding the razor with an open hand.
Sadly, I don't yet have a stone finer than 8000 grit, nor a strop. Just haven't taken that financial plunge yet! In lieu of a traditional leather strop though, I improvised one out of a 2x4, denim, and with some baby powder as a makeshift stropping compound:
Several light passes over the strop and the razor is good enough to use on my whiskers!