Hello phillc!
A very warm welcome to Austria and our little"gun-smoke"-forum here!
Please feel free to bring up all the questions related to your hunting and sports-shooting intentions and do not hesitate to address us in English!
Anyway, congratulations on your impressive deer stalking success in the Highlands!
Or as we would put it in German: "Weidmanns Heil!"
For the reason of being from Tirol (Western Austria) I'm sorry I can't be much of help concerning support in the region of Graz.
But I'm sure you will find someone else amongst our members here.
In regard to your ambitions to go hunting in Austria I suggest you better finish your training in the U.K. and check if that is valid to work as a hunting licence in Austria rather than doing the full training here. If your qualification fits the requirements you might even apply for an Austrian "Jagdkarte"(hunting licence) without doing the training but that is something you will have to check with the guys from the Styrian hunting-federation:
http://www.jagd-stmk.at/ueber-uns/die-s ... gerschaft/If you simply want to get a rifle and go shooting at a range (preparing yourself for the practical test in the U.K.) just hit a local gun shop and find out what you would like to buy. If the rifle is available in the shop and does not have to be ordered it takes a minimum of three days before you are allowed to take the gun home. The rifle must be registered which is a service the gunsmith will do for you.
(Have you thought of what rifle you would like to get yet?

If not you can get some excellent advice from the forks here and they can also help you to find a good dealer/gunsmith in Graz/Styria)
As for the requirements to own and keep a rifle in your house: The authorities will not visit you to check if you have a gun-safe or closet as long as you own hunting rifles -Category C and D- only (and do not do anything irresponsible or silly (like scaring off your neighbours…)
As a rule of thumb: You are fully responsible that nobody has access to your guns, ammunition etc. that is not allowed to have a gun himself. That is especially relevant for children and people under the age of 18, or people that have lost the right to have firearms by having done something irresponsible or illegal, ... Furthermore you must not carry around your guns unless it is for the purpose of going to the range or the gunsmith's and you also must not leave them in your car or somewhere unattended...
So it is best if you keep them in a lockable closet or better gun-safe that will guarantee that you have taken care of your legal obligation to keep your guns out of reach from not qualified people. (Visitors, craftsmen working in your house,...)
Firearms are divided in 4 categories:
Category A: full-auto-rifles, war material etc. (generally not allowed)
Category B: pistols, revolvers, semi-auto-rifles and semi-auto-shotguns (you need a special licence called "Waffenbesitzkarte" to get these firearms)
Category C: hunting rifles with rifled barrels like bolt-action rifles, combined rifles with a minimum of one rifled barrel,...
Category D: shotguns with smoothbore barrels like classical side by side or over&under double-shotguns,...
If you want to bring your rifle to an other European country you will have to get a European Firearms Pass and register the rifle you want to take with you in that document. You might still have to get an official invitation for hunting or a sports-shooting contest to be allowed to bring that rifle with you. The regulations and requirements differ between countries.
Hope this is of help to you.
All the best!
birdofprey
"There is a beast in ever man and it stirs when you put a sword in his hand ..." GoT