Listen to all of the wonderful technique advice above, but don't fall into the trap of "maybe I'm using a bad blade". Or razor for that matter.
Blades make such minimal difference it won't be that.
I have to disagree with that.
I can't use Wilkinson sword blades at all. They cut me to pieces. However, I love Personna Blues. they really suit my skin. YMMV
As to advice to the OP
Don't get caught up in the "More expensive = better" nonsense spouted in forums,
My most exensive soap (DR Harris) causes me to have the horrors. I can't use it without it causing blade rash. My cheapest soap (a tube of Arco) is an utter dream to use and works utterly fantastic.
My most expensive razor (Vintage gillette) doesn't suit me and i get nicks. My cheapest razor (Yuma) suits me wonderfully and is a dream to use
Don't use thick lather. Make it thin. Lather is there to hydrate the hair and lube the razor, anything else is just trying to make amends for poor technique. All soaps are not the same (Well, the good ones are good and the bad ones are bad tbh).
Try some different blades. Blades make the shave. (Astras cut me up as well). Don't try to stretch the life of the blade. Use more than once (some blades get better) but not more than 3-4 times.
Prep: Use proper bar soap (not dove ffs) to clean your face before lathering. Try your stubble before and after a soap wash, you will be amazed.
Map your hair. 1st pass: with 2nd pass: Across 3rd pass against. In reality, whilst its just up-down-across on the 3 passes, you need to not be going against the grain on the 1st pass if you can help it. The real benefits of grain mapping are on your touch ups and allow you to really focus on the final shave pass without overdoing the strokes of the razor. remember it is about reduction, not immediate removal.
rinse after every pass.
Use a good post shave routine.
The biggest effect on myself with regards to not cutting myself was changing my razor to a lightweight razor. I find heavy razors lack decent feedback and i end up over/under shaving.