303 stainless, any austenitic stainless steel is only minimally affected by heat.
The biggest problem is it work hardening on you - from pressure in the cut. This pressure comes from negative rake tooling, dull tooling, too small of a chipload, or too small of a DOC.
Cut it slowly, with a positive rake tool. Keep the chipload up (.004 minimum with an inserted cutter). Make sure the solid carbide tooling and HSS drills are sharp. Leave enough stock for a finish cut to take a decent chip - at least .005/side. When turning it, a common problem is the tool rubbing and causing work hardening in a semifinish cut or using a negative tool for roughing that leaves a work hardened surface behind.
Look for flank wear on the insert. If you see it, change the insert right away. If it is showing flank wear quickly, increase your chipload (as you would any time you see excessive flank wear).
If you keep the SFM down, the IPR/IPT up, and coolant in the cut, then it machines pretty well.
Apparently I don't know anything, so please verify my suggestions with my wife.