Firstly, check out Annies blog! It has insider info on loads of nice places to go: http://irishamericanbreakfast.wordpress.com/
We would recommend just walking everywhere as much as possible and getting lost and asking for directions from locals.
Guinness Storehouse (it really is pretty great - Website - Map). If you are going to or from Guinness, we would recommend walking up or down Meath Street (Map). It's a very local, old fashioned market street in a very old neighbourhood of Dublin. Francis Street is full of old antique stores, and at the bottom of Francis Street is Fallon's Pub (Map), one of our favourites. Also check out Kilmainham Gaol (a perfect history lesson/fascinating tour - Website - Map), The National Gallery (Website - Map). If you get a Hop-on-Hop-off bus ticket, you can do Guinness and Kilmainham in one day (they are near each other) for a better price probably.
If the weather is nice, we would HIGHLY recommend getting the DART (a commuter rail train that runs along Dublin Bay) to either the North (Howth) or the South (Bray/Greystones) and going for a walk. There is a cliff walk from Bray--Greystones (Map) that is fab and takes a few hours. Greystones is a sweet little place to end up. Howth (Map) is a fishing port in the north and there are many different walks around Howth Head (the mountain sticking out into the Bay). The DART to either place is about 45 mins and the walks are a few hours.
The Chester Beatty Library (Website) at Dublin Castle is really a treat and the restaurant/cafe there has fantastic food. If you are into James Joyce, there are readings of his various novels every day in Sweny's pharmacy behind Trinity. (Website - Map)
The Jameson Distillery (Website - Map) is not a working distillery anymore (all Jameson is made in Cork now), so unless you are totally into Whisky, maybe skip it. You could still go down for a cocktail and tasting there without doing the tour. The Natural History Museum (Website - Map) and The National Gallery (Website - Map) are free and both have some fun things to look at if the weather is not nice. It's worth walking around in/through Trinity College (Website), but unless you are HUGE illuminated manuscript fans, skip paying in to see the Book of Kells (Website). However, seeing the Long Room (the original big library) is pretty great. You can stick your heads into the Chapel in Trinity if it's open and all of the buildings are open and free to anyone who wants to poke in usually.
If you want to go to the movies, The Lighthouse Cinema (Website - Map) in Smithfield Market is the one to go to. Smithfield is a totally different feeling sort of part of the city. The Black Sheep (Website)--a pub with many craft beers--is near there as well and worth a visit.