Read the Home Office guidance notes - they are on a sticky at top of the Forum - this is what the Police FEO's should be working to.
For Deer Stalking you need a Deer Legal Calibre - slights differences between Scotland and England, but fundamentally a 243 with a 100gn bullet at c 2,900 fps is the minimum for all deer in the UK. Yes there are lots and lots of deer legal calibres, but the vast majority of stalkers use a 243, a 308 or a 270 for the simple reason that ammo is cheap and readily available in just about any gunshop. 243 tends to be the first choice for smaller deer, 270 win is preferred on Scottish Hill for Red, and 308 is now the standard issue for Forestry Commission Rangers and seems to be first choice for many. it used to be £1 a bang for deer cartridges. It's getting closer to £2 a bang.
Gallery rifles are a whole different ballgame, which I have no experience, but have little to do with Deer Stalking, and certainly for Scotland with min Muzzle Velocity requirements of 2,450 fps are not deer legal.
To get an FAC, it's not, thankfully, just a question of ticking the boxes, it's much much more a question of attitude and responsibility. To get an FAC and the authorisation to use a deer legal calibre on your own is even harder.
The best advice for somebody who wants to go deer stalking, is to find somebody who is a guide and who will take you out and you can use their rifle and get experience. If you want to shoot Gallery Rifles - then join a shooting club and get experience that way. But being in a target club is not automatic right to get a stalking rifle.
Shooting of any form can be expensive unless you are being paid to do it, in which case it then becomes your livelihood. The Target Boys can shoot thousands of rounds a year and that's a significant expense in terms of not only ammo, but barrels as well. With Deer Stalking - ammo is the least of your costs.
Any hobby, pastime costs money. Any hobby using expensive equipment, access to land etc etc - even more. But if you stalk regularly you don't a subscription to Satellite TV, Gym, Golf Club etc etc etc. Keep some spare for bribes or a divorce lawyer though