On the northside you could consider:
Malahide (Grand Hotel) -seaside town, train line to city centre, nice restaurants
Drumcondra - (Bonnington hotel, Skylon hotel) closer to city, regular buses, quite studenty some restaurants (shouk as previously mentioned is good for vegetarian)
Southside you could look at the Clayton Hotel Leopardstown which is along the luas (tram) line or Marine Hotel Dun Laoghaire for the train.
These are just a few options but might get you started.
In terms of what to do you could explore the area around Trinity- Grafton St, St. Stephens Green etc. Cornucopia is a nice vegetarian restaurant near Trinity.
You could go to Rathmines ( a studenty area) where you could go to the lovely Stella cinema and eat in Bethlehem restaurant.
Kilmainham jail or EPIC museum would be good for a bit of history.
Botanic Gardens, Phoenix Park, Malahide Castle, Sandymount Strand all nice for walks.
Howth is a nice village to visit.
Brother Hubbard is nice for vegetarian brunch. They have a few locations. Nutbutter on Grand Canal Dock is a nice vegetarian option. The Bord Gais theatre is close by if you are interested in that.
Almost all restaurants have vegetarian options so you would not have to worry about sticking to only vegetarian ones.
You will find Dublin quite expensive compared to the UK.