Double edge safety razor for me.
I have a couple of different razors-I started with an old Gillette that was about $10, but now mostly use my grandfather's old Gillette Tech. Old Gillettes are usually cheap to buy and for the simpler ones there really is nothing to go wrong(I have a Slim that unfortunately is way out of adjustment). I was on vacation and in a pinch, so went to Walgreens and bought a new Van Der Hagen branded razor-it looks to be more or less a copy of a basic twist to open Gillette and is not bad at all.
I sampled the full gamut of blades, but settled on Derby blades changed every 3 days. They are about $15 for 200 on Amazon. I've also come to appreciate Feather blades as I've become more experienced, but don't use them that often. Even if I did, though, they're still cheaper than cartridge razors at $25/100.
I rarely cut myself, especially now that I've been using one for ~2 years. For the times I do, I keep a can of grocery store Alum(from the spice rack) by my sink. I just wet my finger, sprinkle on a little bit to make a paste, and rub it over the cut-that will stop it bleeding immediately(alum is what styptic pencils use).