Hi Kevin

If the bearings are just standard not too expensive to replace angular contact bearings I would use run of the mill light duty bearing grease, but not too much no more than 30% of the space in the bearing should be filled. Otherwise you are likely to get overheating.

I would also start the machine at the lowest speed and let it run for 15 – 30 minutes, then change to the next speed up and so on. If the spindle overheats the preload may be too high or you put in too much or the wrong grease.

There is a good discussion on different types and brands here:
Beware some specialty greases are quite expensive.

**LINK**

Regards
john

Overheating of the bearing due to overpacking with grease will only really happen if the bearing is run at near its maximum speed. The speeds the bearings run in a milling spindle will not overheat if overpacked. The only likely hood of overheating is if the preload is too high.