its going to cost you a lot to get started.
Do not get a 125. You will grow out of a 125 in 6 months and overtaking anything on something that underpowered is dangerous. you will need 250 minimum for a 25 mile commute. I started with a 250 virago for a 20 miler. It went 70 mph max on the motorway and done the job for a while. 400 cc now and hoping to upgrade soon.
You will need to apply for an A licence and not an A1 and yes you can go out on the provisional.
By law you are restricted to a low powered bike up until you pass your test and then for 2 years. You can of course buy any bike you want and get a restrictor kit fitted but I would recommend just getting a 250 to start with.
You will need a helmet, boots, good gloves, leggings and a jacket. They are not cheap but get the best you can afford.
A second hand 250 will set you back around 2.5 to 3 grand i reckon. Check cbg.ie used bikes section for ideas.
You are going to get screwed with insurance and there is a choice of 3 expensive companies where you can get cover.
Bikes are great though, i have the bug now after a few years of it. Good luck!