Have you a particular firearm in mind and what type of vermin _ quarry will you be after?

Since you don't inculde birds on your quarry list why not consider a 17HMR. The 17 will knock foxs at a better range then the .22 will and perfect for rabbits and birds (Corvids etc) on the ground.

Pros for the 17HMR, modern guns and ammo are inherently accurate. Flat shooting round out to sensible ranges, so you can shoot rabbits near and far without thinking holdover or dialing. The little .17 disintegrates when it hits the ground lessing the chance of ricochets. Cons, wind deflection is its biggest enemy. Compared to a .22 the ammo is dearer. The rifle itself can be dearer then the average secound hand .22.

Pros for .22LR, one of the most common rifle/ ammunition combination in the world. Most decent rifles and ammunition will give you realistic field/hunting accuracy out of the box or off the shelf. Because they are common, good quailty rifles are plentiful and cheap to buy. Assessories such as scopes, moderators and bipods dont have to cost a fortune to put together a good shooting package. Ammunition is cheep and plentiful. Cons, realisticly the humble .22LR low velocity has range of 65/75 yards before the trajectory of the round requires serious over holding or dialing of the scope. Plenty of foxs are humanly killed with .22 but they need to be in close to optimise the little bullets energy. The .22 is prone to ricochets, so care must be taken with it .

Pros for Shotguns. You can spend any amount you want, cheap single barrel guns can be picked up for €50. Ammo is relatively cheap. They are versatile allowing hunting of birds, rabbits and foxs. They can be considered inherently safer to use as the range is short. Depending on the type (SxS, O/U, Semi Auto, Pump) they are simple to use and maintain. Cons You can spend any amount you want, guns can go into the tens of thousands of euro. There range is short, so limiting you to certain hunting styles and conditions. Depending on the type (SxS, O/U, Semi Auto, Pump) they can be complicated to use and maintain.

Just Edited, ment to add 22WMR, this older round has had a make over recently enough, making it a very versatile hunting package. Ammo is dearer then 22LR and on par with .17 cost. A lot of older guns around that would be worth a look as the round fell out of favour with the introduction of the .17HMR and the .223, but still a good one to look at for fox and rabbits.

Ideally get out with someone and have a go of a few various firearms.

In theory there's nothing to stop you from applying for any legal firearm for your first application, once you can prove just cause to have it licensed. So if you want a .223, knock yourself out, perfect for foxing and reasonable for rabbits.

There is no legal stance as to the content, syllabi or duration of a firearms compentcy course, so if a firearms dealer takes you in the back room and spends 10 minutes discussing the ins and outs of a particular firearm and relevant safety, then gives you a certificate well hey your covered and the box is ticked. But if you have no experience with firearms then really for your own sake and the sake of others get yourself a good course. Contact the NARGC or one of the professional organisations that are providing firearms training covering safety practices, operation/function, cleaning and some provide basic practical marksmanship principles.