If you want to avoid Dublin, Cork would seem an obvious choice. It's the second largest town, has a good reputation and is located in a picturesque region (I don't have good memories of Limerick but I don't know it well). But many small to medium-sized towns are pleasant enough (I lived in one of them some years ago and remember it fondly). Places like Galway, Waterford or even smaller county towns like Sligo could be perfectly fine.
The thing is, apart from Dublin, none of them will have a real city feeling. So if you are used to live in a place with many cultural venues, clubs, shopping opportunities, etc., don't care for outdoors activities, and are really concerned about getting bored, you really have no other choice than Dublin. Also, when I was there, Dublin was really expensive but other cities weren't cheap either.
As far as foreigners are concerned, you will find some everywhere but mostly ERASMUS students (quite a lot of those as Ireland is nice and English-speaking so very attractive for German, French, Spanish, etc. who want to get an experience abroad) and tourists from Europe and North America. Apart from that, the country is not very diverse. There are probably more expats in the Dublin area but don't expect it to be like Paris, London or Amsterdam either.
Finally, transportation is a bit slow by European standards. Most flights out of the country depart from Dublin and getting there from the west coast takes several hours. So if you are travelling frequently or like week-end breaks abroad, living outside Dublin will add significant transit times.