Dublin and Belfast have comprehensive local bus networks, as do some other larger towns.
The big decision in getting around Ireland is whether to go by car or use public transport. Your own car will make the best use of your time and help you reach even the most remote of places. It's usually easy to get very cheap rentals – €10 per day or less is common – and if two or more are travelling together, the fee for rental and petrol can be cheaper than bus fares.
The bus network, made up of a mix of public and private operators, is extensive and generally quite competitive – although journey times can be slow and lots of the points of interest outside towns are not served. The rail network is quicker but more limited, serving only some major towns and cities. Both buses and trains get busy during peak times; you'll need to book in advance to be guaranteed a seat.
Border crossings between Northern Ireland and the Republic are unnoticeable; there are no formalities of any kind. However, this may change once Brexit occurs in 2019.