For this itinerary, it's best to rent a car so you have the flexibility to explore each location thoroughly. Expect to spend roughly €30-50 per day on car rental. Keep in mind that driving in Ireland can be challenging, especially if you're not used to driving on the left side of the road. Alternatively, you could book a private tour instead, which will take care of transportation and give you the added benefit of having a knowledgeable guide.
If you have more time, consider adding a day trip to the Aran Islands or visiting the Ring of Kerry. If you have less time, consider skipping Belfast and spending an extra day in Dublin or Cork. Don't forget to pack layers - the weather in Ireland can be unpredictable!