Very impressive!
What is the procedure for purchasing (respectively) a rifle, a shotgun and a pistol? Does one have to deal with a lot of bureaucracy to do so?
Yes.
It differs somewhat from state to state, and I can only really speak for my own.
What we have is a graded licencing system. A basic A/B licence that allows you to own basic hunting style rifles (non-semi automatic) and shotguns is quite easy to get as long as you don't have a criminal record. But there's a 28 day waiting period for a first licence.
And even with a licence you need a permit to purchase any gun - and that takes about ten days to get. It's pointless paperwork and it's the part of the process I object to (I personally don't have a problem with requiring a gun licence, viewing it as similar to a drivers licence - but once you have your drivers licence you can just buy a car. Not that easy with guns.
If you want a handgun or a semi-automatic rifle/shotgun, you need a more specialised licence (Category H for handguns, Category C for the longarms) and that requires forms and an interview to get and can be refused for even pretty minor reasons. There's appeals processes if refused, but that just makes it even more complex.
Anything higher than that - fully automatics etc, are theoretically possible but in practice, most people won't get a licence for those.