In February I'd recommend layers. It can really vary between freezing cold torrential rain or a crisp sunny spring day. Don't make the mistake of wearing rain gear, you'll stick out like a sore thumb. It rains alot but unless your climbing a mountain a decent coat and an umbrella is the norm, not full gor tex rain gear.
We can give you better advise if you let us know what you usually do on holidays and how long it's going to be. For example, if you like museums I'd recommend collins barracks over the national history museum. (BTW alot of the bigger museums are free). The Guinness brewery is the most popular attraction in Dublin but my parents went to the Jameson brewery in smithfield last week and loved it. I'd also highly recommend the little museum of dublin, especially the restaurant.
Keep in mind, Ireland doesn't really have off season but some of the less popular castles and manor houses may be closed for february but if the weather is good they may reopen. Don't make the mistake of thinking it'll be quiet as some Irish people have mini holidays in february especially around the big tourist areas like Donegal, Kerry and Galway. Actually that reminds me TedFest is on in February in Galway. It's a festival in celebration of a comedy programme called Fr Ted.
You can drive across the country in 8 hours so it's not as if your stuck in the one area. On the east coast Dublin, Wicklow and Kilkenny would be the big tourist centres while on the west coast they're probably Cork, Kerry and Galway.
In the cities public transport is a must but for everywhere else you're going to need a car. For Dublin Bus you'll need correct change, they won't accept notes/papermoney.
I live in Dublin so if you have any specific questions like where to eat etc let me know. I'd also know general information about other areas. If your travelling up to Belfast I'd recommend stopping in Carlingford, it's a beautiful village and one of the few fjords in europe. It's in the cooley mountains which is where Cu Cuchulain and the tain were located and is full of great scenery and folklore.
Keep in mind our cities would be the equivalent of an American town so not very large. Taxi-drivers are often asked by tourists to take them to the city centre when they're already in it. So while you've said you don't like urban sprawl I wouldn't discount it in Ireland.