As most others have said - get a hybrid
Tyre thickness should be between that of a mountain bike and a racer. Look for extra thick tyres that are puncture resistant. If they don't come with the bike, you can fit them as extras.
And make sure it has mudguards. Plenty of crap on Dublin roads even when it isn't raining.
Also make sure it has a carrier, as implied by those who've recommended panniers.
I'll probably jinx myself by saying this, but I bought a Dawes hybrid about 6 years ago. It does loads of commuting as well as some touring in Ireland and abroad. I've never got a puncture. I've changed the tyres once due to wear. When I changed them I got ones with an extra thick lining to be puncture-resistant.
Regards carrying about all the tools etc - don't bother. The only thing I'd consider carrying when commuting is a pump and wet gear (obviously carry more when touring). If you look after the bike reasonably well at home, the chances of something happening on the side of the road that need immediate repair are low. Worst comes to the worst, walk to the nearest bike shop, or lock the bike and hail a cab. Pick up bike later to get it fixed (or put it into boot of cab there and then). Less hassle in the long run IMHO.
But if you really want to carry tools, get three tyre levers and a "multi-tool" from a bike shop - small, compact and contains everything you need (even a chainbreaker).
Regarding shops, commuting solutions in Rathmines, or Cycleways on Wexford/Camden st, or the one in Parnell st previously mentioned. You'll pick up something decent enough for €350, though could easily spend more.
Sensible idea to build yourself up to it by cycling evenings first. Good luck with it - it's the best form of transport in rush-hour Dublin.