Ireland/Northern Ireland in 8 days

Rick has an 8 day tour of Ireland (possibly starting in August depending on Covid) that comes pretty close to what you what you are looking for. It can give you an idea of time and costs of accommodation and transportation. You might be able to do the trip for less money but you would have to be ready to make all your arrangements a couple months ahead of time. He also has a 14 day trip. It would be worth it to extend your trip.

Assuming you are travelling from North America? Then you would only have 6 days of travel time actually in Ireland. I think your plans are overly optimistic. A thorough reading of a few guidebooks to nail down exactly what you want to see in each area, plus investigating travel modes and times between each area should help you pare down your itinerary.

There are no nonstop flights from between the US and Belfast, but you can fly nonstop between: Atlanta, Boston, Chicago, Los Angeles, New York, Philadelphia, San Francisco and Washington DC to or from Dublin. Since your time is limited, I would fly into Dublin and out of Shannon (or vice vs.) You can also fly nonstop between Shannon to or from Boston and New York.
I would arrive or depart from Dublin and spend a minimum of three nights there and make day two a day trip to Belfast via train. I would then take a train to Galway from Dublin and sleep there one night before taking a bus to Rossaveal to spend one night on Inishmore. I would then rent a car in Galway and drive to Dingle spending two nights there before dropping the car off at the Shannon airport.

Are you driving? This is not possible by public transportation. As said you only have at most eight days if you hit the ground running. The arrival day is only about a half a day by the time you leave the airport get the car etc. The departure day usually means a morning flight to the US and with US preclearance you need a good three hours in the airport before your flight leaves for home. That leaves you six and a half days total. I would definitely leave out the west coast and Dingle and the Aran islands they are just too far and driving there is much much slower. If this is my trip I would focus on just Dublin for a couple of days Belfast for a couple then perhaps Galway and home.

You say you have 8 days, but what does that mean? How many nights do you have. We had 7 nights in Ireland. Upon arrival in Dublin we got a car and drove directly to Kilkenny. I would not have wanted to drive further on arrival day from the west coast of the U.S. We had lunch, toured the castle (I remember little of it) and walked around before checking into our B&B. After dinner we crashed hard. In the morning we departed, stopping at the Rock of Cashel on our way to a 2 night stay in Dingle. Loved that area. We then drove north. Stopping at the Cliffs of Moher on our way to Galway for a night. We didn’t have much time for Galway unfortunately. We then drove back to Dublin and returned the car for a 3 night stay there. On our last morning we had to go directly to the airport. It was fast paced, but we enjoyed the time we had and are hoping for a return trip next year. If Belfast is a must you might do better to stay towards the eastern side to maximize your time.

As others have said you really can't cover all of the territory that you're proposing in the amount of time you have available. Fact is, you can't even come close unless you want to spend most of your time in a car or on public transport just getting from place to place.
It would be easier logistically, especially if you'd rather not rent a car, to base yourself in Dublin for several days and use it as a base for easy day trips by rail or bus north to Belfast (about 2 hours) or south to Kilkenny (maybe an hour and a half).
Galway would be reachable in about 2 hours by train. From there you could book your trip out to the Aran Islands either via the Rossaveal Ferry or by air. The short flight isn't that much more expensive than the ferry, and gets you there and back much quicker and without the worry of a rough crossing or outright cancellation in bad weather or high sea states. There are day tours available in Galway to give you an overview of Connemara ... probably the best use of your limited time rather than trying to do it independently.
Since airfares are much cheaper flying in and out of Dublin, you'd save money by returning there the night before your return flight.
Trying to include Kerry & Dingle would be too much of a stretch I think.

I don't think that this is realistic. You will need to edit your trip and decide what is the most important to you and what you will leave for the next trip to Ireland.
Day1/2 If you are a west coaster you leave US on one day; arrive on day 2 in Ireland. You will probably arrive in Dublin.
Spend night 2 and 3 in Dublin.
Spend night 4 in Belfast
Now you have just a short period of time left Nights 5 and 6 before you return to Dublin for night 7 before you fly out (assumes RT in and out of Dublin)

A few options:

You can focus on Northern Ireland and spend nights 5,6 up on the Antrim Coast (one of my favorite spots) This is a particularly good idea if you are hoping to do your trip by public transport as these connections are convenient.

You can focus on the Galway area. Night 5 in Galway and spend day 6 out at the Aran Islands before returning for night 6 in Galway. Easy to return to Dublin and the airport from there. This option is also easily done with public transport.

You can focus on Dingle and spend nights 5 and 6 there. No good public transport options so this would require renting a car in order to cover enough territory.

You can have a great trip in 8 days. You will need to make some choices on where to put your focus. Always give yourself permission to come back another time.

Drop Dingle, Kilkenny and Dublin. Belfast is more interesting than Dublin IMHO.

Fly to Belfast, spend a day in Belfast, (Titanic, take a political tour), take a couple of days to drive from Belfast to Derry, a day in Derry, take 2 days to drive from Derry through Connemara to Galway, and spend a day on an Arran Island, and a day in Galway. Get to Shannon and fly home.
Dublin is OK, Kilkenny is wonderful, Dingle is great once you leave the tourist trap that is DingleTown.
But you won't be happy trying to do it all in 8 days.
We went in September 2019; it is a great time of year.