Annealing is a slow process used to soften a steel to an "easy" to work condition. Often the steel we buy is already annealed.
Normalizing is when you heat it up to let the steel relax and release any stress that built up from getting shipped all over the place. It's not easy being steel ya know. Rough life with all that pounding and pouring... Seriously though, normalizing is a stress relieving process that also helps make sure you don't get cracks or warping during HT. I normalize after profiling and mostly flat grinding the blank. Then I finish flattening it and grind the blade. I normalize again prior to HT, basically as part of the process, checking before doing the actual HT that it's still looking right.
I don't think I really need to do both, probably normalizing before HT would be sufficient, but I haven't done enough to test that theory.
Spheroidizing is annealing on steroids and makes for even softer and more ductile material. In practice, when tempering we are doing a very limited form of this to reduce the hardness of the steel to a practical level and relieve stresses that built up during quench.
If I've got any of this wrong I'm sure one of the more experienced folks will come educate both of us.