When purchasing your equipment for alpine climbing, what is the order that you should purchase your gear starting out? Is there an established protocol as there is with backpacking?

First off, "alpine climbing" is really broad. Do you mean technical ascents (that is, with a rope and belays), or non-technical ascents where ice axe and crampons will suffice?

What experience (and gear) do you have? Sport climbing, hiking, backpacking, etc?

Before purchasing equipment..especially with the alpine climbing, is there someone who will show you how to use it (ice axes, crampons, rope management, rock gear and gear placements, etc.)?

The established protocol is to go out and do what you can with what you have. As you get more technical you will become more aware of what you need. Don't buy anything until you have found you need it.

Strap on crampons and a basic ice ax first. All you need for the first few and from there it is really easy to figure out what you need.

Currently I'm looking to get my start by doing guided trips/classes. I'm looking to do a winter Presidential's Traverse and I want to do some ice climbing as well. That will probably be about it for this winter for me. I currently go backpacking regularly and plan to go to the climbing gym to get some more experience there as well. I figure I will be starting out non-technical, but moving towards technical after I gain some more experience/comfort or find someone I can tag along with.

I gather that crampons and ice axe might be the first to look at then? I was thinking boots too for the comfort factor.

Thanks for the help/advice so far.

Your guide service should give you a list of everything you need for your trip/course. Start out by buying only what you need for each course you take.