The main thing to remember that a lot of people do they think more revs equals cutting. A good feedrate with plenty of coolant or compound and the right revs will be ok. Especially with stainless if you let the drill rub without cutting you will blue the drill bugering it and work harden the surface which will be an absolute ****** to get through with even a carbide drill.
You probably already one- a bench grinder. There's various guides and vids on the interweb and it takes a little practice to get results that aren't wonky
Too fast, yes. Too much pressure, not so much- as said if the feed/pressure isn't kept up then the drill rubs instead of cutting and then the stainless workhardens, the drill overheats and it all goes wrong.
The Dormer A002 is what I use and I work in stainless most if the time, it's a split point type, always use the correct speed and constant pressure on stainless otherwise you will blunt the drill cutting edge.
Don't centre punch stainless either it just hardens it.
We run cheapie HSCO (high speed steel+ Cobalt 5% ish) Made by COMAC. Only on smaller sizes sort of 3-9mm on the CNC mills and lathes and it will run nearly 200 parts before even rubbing the coating off on 316 stainless in a 40mm deep hole!