It's safe to walk around Dublin city centre, and you can almost get to everywhere within 30 mins walk. The Temple Bar area will be quite busy during the weekend.
There are buses and Luas (local tram) running arcoss Dublin. You can get a Leap card for traveling on them at many convinent shops around Dublin. For more information about the local buses: https://www.citizensinformation.ie/en/travel_and_recreation/public_transport/bus_services_in_ireland.html
Taxi in Dublin comes in various colors, but they all have an identifiable sign saying "Taxi" on top. A lot of taxi in Dublin is cash only, but you can also call Taxi via app like Uber or FreeNow (which you can pay via the app).
Free Now is recommended taxi app: https://free-now.com/ie/
Uber is available in Dublin: https://www.uber.com/global/en/cities/dublin/