There are many ways to remove worn or damaged bearings from shafts and housings. I'm not bragging or confessing, but here are some methods that have worked for me:
-Hammer and chisel/punch. If you can get behind a bearing and use a hammer and punch to drive the old bearing off the shaft, you're having a good day. No muss, no fuss, and as long as you don't mar the shaft, installing a new bearing is a breeze. Use safety glasses--bearing races are hardened metal that often shatter or chip when struck.
-Air hammer and pry bar. This is the technique I usually try first because if it works, it's fast and clean. Run an air hammer against the end of the shaft while prying on the bearing or its housing. Make sure you've cleaned all paint and corrosion off the end of the shaft, and the buzzing action of the air hammer will often slide the bearing smoothly off the shaft. Use safety glasses and wear gloves to prevent injuries from shattered bearing housings.
-Cut-off wheel/die grinder. If the bearing is exposed you can use a die grinder with either a cut-off wheel or a cutting tip to surgically cut through the outer and inner races. It takes time, but is a very "clean" way to remove bearings. Again, use safety glasses.
-Gear pullers and pry bars. If you have room to get the jaws of a gear puller behind a bearing or its housing, a gear puller is another clean alternative that's fairly quick. Bearing splitters and other special tools are nice if you can get them into place and IF you have the right tool for the job.
--Acetylene torch. Okay, long-time readers knew I would get to this one eventually. Burning off bearings with a torch is crude, dirty and potentially dangerous. But it is the fastest way to remove a bearing I've found. I always have a fire extinguisher on hand and liberally dampen with water the area adjacent to the bearing I'm about to destroy. There will be flames, sometimes a LOT of flames, when bearing grease ignites. There will be lots of slag and sparks and small pieces falling and flying around. There is the potential to damage shafts, housings and wiring harnesses in the vicinity of the bearing being removed. It takes practice to remove bearings without nicking the shaft or bearing housing. But it can be done, and it's fast and functional when done with care and caution. When possible, I try to let shafts and housings cool before I install new bearings, otherwise the hot components can melt the "factory grease" out of the new bearing.
There are dozens of other ways to remove bearings. The "right way" is the way that allows you to get the job done comfortably, safely and as fast as possible. For me, I always try to buzz a bearing off with an air hammer and pry bar, then... get out the garden hose and keep it handy because there are going to be lots of sparks and flames.