My dermatologist "yelled" at me for using a multi-blade razor a few weeks ago. He was able to tell that I did based on bumps I had on my neck. My ingrown hair or other issues are never that bad(well they were when I was using electric) but I guess it was enough to notice. I am using the Hydro 5 with Neutrogena Shave Gel.
Now I'm looking into alternative methods not even so much for the razor bumps/ingrown hair, which isn't that bad, but probably more for the expense factor of using cartridges. If I actually keep up on shaving, I will shave 2-4 times a week which would make me go through a cartridge probably once every two weeks if not a little sooner. Then it's about $12-$15 for a 4 cartridge refill. I don't like the hair getting caught in between blades too much either and it seems like I have to use too many passes sometimes to get a closer shave the more and more I use it. Maybe I should be replacing those blades faster than I am.
I'm considering going to a DE safety razor with the brush, soap, a pack of 10 razors to start and after shave cleanser/moisturizer. Pretty much going with a standard kit that others have already posted which is the Merkur Model 180 long handle, badger brush, stand, razors and Proraso sensitive soap. I think it would be cool for the experience itself and the longer term cost savings. I'm just scared of cutting myself, which I currently rarely (if ever) do. So, I'll probably invest in one of those pen things to start too but hopefully I won't have to use it often...
You'll want a styptic pencil. They're only a couple of bucks and last forever. Another option is an alum block, bit larger, more expensive and last forever as well. Both work well at staunching the bleeding from small cuts (weepers.) The alum block has the added benefit of acting as a general skin balm.
With practice, weepers become a rarity. You may occasionally get a small cut, though, from carelessness or even a dulled blade. In such a case, a styptic pencil is a lifesaver.
I would recommend getting a variety pack of blades only because blades aren't a one-size-fits-all item. Take a Derby blade. Some people love them. I may as well be shaving with dull glass when I use a Derby.
If I had to choose one blade to recommend, it would be Feathers and that is only because the have a reputation for being sharp. Some people have less than stellar experiences with Feathers, just like any other blade. Those, though, tend to think that the Feather is just too sharp. Hardly the worst thing someone could say about a blade.
Again, my first recommendation would be to plonk down the 20-30 on a variety pack. You'll almost assuredly find a blade that works for you in a pack. If you buy just one type of blade and have a bad experience, it could sour you on safety razors altogether and that would be a shame. In addition to the all-around better shaving experience, the cost is so much less once you have your kit worked out.