Dublin city centre is well serviced by public transport - a combination of buses, trains and trams. Dublin Airport does not have a direct train or tram link. There are several bus and taxi options. https://www.dublinpublictransport.ie/busaras
Bikes and electric bikes are available here https://www.dublincity.ie/sports-and-healthy-living/walking-and-cycling-dublin/bike-sharing-services-dublin however Dublin has quite poor bike infrastructure, some crazy driving, and if you are not used to the city, is perhaps best avoided.
Leap Cards
For the bus, train and tram you can Tap-On and Tap-Off public transport if you have a Leap card. You can purchase a Visitor Leap Card in advance https://about.leapcard.ie/leap-visitor-card but this will be posted to you.
You can purchase a standard Leap card at selected agents in Dublin Airport, shops in Dublin city centre, or directly from a ticket machine at a DART station - or buy direct at the machine in the station
From or to Dublin Airport
Dublin Airport is north of Dublin City Centre and approx a 20 minute drive to the city centre (without traffic). Car hire options are here https://www.dublinairport.com/to-from-the-airport/car-rental. Be careful exiting Dublin Airport - the road connects very quickly to the motorway. The Dublin Port tunnel, can reduce travel time to the city centre, there is a toll charge.
By Bus
There are several bus transfer options https://www.dublinairport.com/to-from-the-airport/by-bus/dublin-buses including the public bus or private operators.
Aircoach - or known in Dublin as the 'Blue Bus' is very popular https://www.aircoach.ie/. Aircoach is a private operator so Leap Cards are not accepted. You can book your tickets in advance and there is a bus tracker on the website. Whilst you can buy a ticket directly at the airport on arrival, during peak travel season, the blue bus can be sold out so recommended to book if choosing this option.
By Taxi
FreeNow is widely used to order taxis in Ireland, it is similar to Uber. Irish taxis require licensed drivers and therefore Uber has restricted policies and dispatches licensed taxis so it is usually more expensive as you are paying an additional booking fee on top. You can hail a taxi on the street in Ireland - available taxis will have a lighted sign but you can just stick your hand out and they will stop if they are available. There are also on street taxi ranks where taxis will be waiting - these are often located near hotels. Most taxis take cards but you may have to pay cash - you can also request a taxi fare receipt.
About the Train
DART (metro)
Dublin's commuter train (the DART - Dublin Area Rapid Transit) follows the curve of Dublin Bay. The trains will be identifiable by the last station. Trains going North will have a final destination of Malahide or Howth. Trains going South will have a final destination of Bray or Greystones. The closest DART stations to Trinity College are Pearse St or Tara St stations.
Regional Trains
Mainline trains which connect the city to rural areas are serviced by two primary city centre stations which are not connected to each other, these are Heuston Station and Connolly Station. The train operator for all services is Iarnrod Eireann / Irish Rail https://www.irishrail.ie/en-ie/train-timetables/timetables-by-route
If you have mobility issues you will need to check that the station you wish to arrive or depart from has the facililty to board or deboard you - frequently lifts are out of service, or stations are unmanned.
Luas (tram)
Luas is Dublin's light rail or tram system which services the area slightly away from the coastline. It has two lines - the red line going west from Dublin city centre, and the green line which goes from the north to south. Dawson St and Stephen's Green stops are closest to Trinity College Dublin. Information on routes and tickets is here https://www.luas.ie/luas-map/