I live in Dublin. It's quite a small city with decent public transport. There is an electric train that runs along the coast called the DART, also trams in the city called Luas and buses that connect the city with lots of suburbs. We also have city bikes to rent. You need to register on www.dublinbikes.ie
in advance and you can collect and drop off bikes throughout the city. As Emberlyn suggested it might be woth taking a Hop-on Hop-off tour to get your bearings on the first day. There are plenty of Airbnbs in the city that provide budget accomodation or try sites like Trivago or lastminute.com for hotel deals.
A lot of visitors to Dublin like to visit Christ Church Cathedral Guinness Storehouse Trinity College Dublin Castle National Gallery of Ireland Irish Museum of Modern Art Dublin Writers Museum Collins Barracks
I would add Famine Memorial Farmleigh House Glasnevin Cemetery
(with a trip afterwards for a pint of Guinness in John Kavanagh (Grave Diggers)
(Gravediggers' pub) Take a wander around parts of old Dublin like Stoneybatter
and, if you have time take a trip on the DART from Tara Street Train Station
(Southbound) or Howth
(Northbound) to get coastal views and take a walk by the sea. There are wonderful views of Dublin from Killiney Hill Park
( a 15 minute walk from Dalkey DART station) or walk along Howth Head or pier and have a good seafood lunch/dinner in either village.
I would skip Temple Bar