You have to be careful. For me, its all about the angle of the blade and how much is exposed. I used a cheap DE razor and it took the face off me - more than likely because I wasnt used to it or or to use it properly.
So, to start with, I would strongly recommend you buy a razor with an adjustable head that you can set how harsh the cut is. Once youve mastered it then you can move to cheaper ones because you will be able to handle it better.
If you go cheap to start it might put you off forever.
I bought a rockwell with 3 different reversible heads giving 6 different levels. I can use 4 for the first shave then maybe 3 for the 2 shaves against the grain. If I'm in a rush, I can do one pass with the 6 and leave it. When I started the 6 left my skin very sore only because I was learning.
Mind you, I have sensitive skin so if you dont you might not have to worry so much.