I just started shaving with a double-edge(DE) safety razor a few weeks ago, and while it seems like much less irritation than my old Fusion cartridges, I think some of the techniques people stress for DE shaving would have also improved shaving with cartridge/disposable razors too.
Using a brush + shaving cream is huge. Massage it in to your face for a good minute or so. When I shaved with my Mach 3 I always just put some shaving cream in my hand, rubbed it on my face real quick, and then shaved. Rubbing the lather onto your face with a brush not only applies it to all the individual hairs better, but works up the lather much better as well and the brush swirling around on your face tends to push your whiskers upright for easier shaving
The second main factor is to figure out which way your hair grows. I started using the technique with my DE of doing one pass of shaving completely WITH the grain of my facial hear, then rather than the second pass being against the grain(irritation), the second pass is at a 90-degree angle to the grain, or across the grain. So basically if shaving north to south is with the grain, and south to north is against the grain, then an east-west motion would be across. Gets you a closer shave than only going with the grain, but doesn't irritate as bad as going completely against it.
A lot of people do a third pass of against the grain at that point, but I find that it still irritates my skin, even though there isn't much left after the first 2 passes, so I just skip that now(plus, who has time for shaving basically 3 times, I'm usually cutting it close as it is with not being late to work)
Also obviously re-lather between each pass, which if you have a brush is no big deal, it'll still be lathered up from the first time when you use it the 2nd.