I use both a DE saftey razor and a disposable straight edge (uses a half of a DE razor blade). The straight uses Shark single edge blades, and the double edge uses the new style (yellow package) Feather razors. I replace them every three shaves. Each time I shave, I shave first with the grain (with straight), then perpendicular to the grain (double), then against the grain (double).
The result is skin so smooth you'd never know I had facial hair and not a single bump or burn. I do have ultra-sensitive skin, though, so I do use some aftershave and lotion.
In contrast, I *always* got razor burn and bumps (and sometimes ingrowns) when I used modern cartridge razors no matter what aftershave products I used. And that was only going with the grain. Against? Ha, it'd do nothing for the hairs but open up wounds all over my face.
You can pick a pack of 100 Feather blades for <$40, and they're usually the most expensive double edge blades you can buy (but also by far the sharpest IMO). If you go with a Shark, Gillette, or Derby double edged blade, you can get 100 for <$20, usually. Plus they often sell variety sample packs for cheap so you can find out what blade works best for you.
TL;DR -> Go double edged saftey and a good soap/brush and your face will thank you.