Car rental

Look at Autoeurope.com, a rental car broker in Maine. They are very helpful if there are any problems. We have used them for many years.

Careful there. You're talking about an international rental for which there will probably be a punishing one-way drop charge. You might be better off taking the train from Belfast to Dublin (about 2 hrs), renting the car there and then dropping it off at Shannon, for which there will probably be no one-way drop penalty.
Many here recommend Dan Dooley - reasonable rates, and they're about the only company that clearly explains the matter of the insurance, which can be difficult to sort out with most of the others.

thank you!

You may have some other issues in addition to the good points made above.

Four adults, generally not a problem. Four adults and all their baggage? That gets trickier. You will need a larger car (how much larger depends on how much baggage and how large the passengers). Larger cars get more expensive, and - depending on where you're planning to drive - in a bigger car, the driving may get more challenging on very narrow lanes. Be sure you factor these in.

As far as cost, you will need to compare different rental companies. He aware that CDW insurance is mandatory in Ireland, and some companies charge out the nose for it. It is the all inclusive prices that matter.
Gasoline/petrol is pushing $6 per U.S. Gallon in Ireland, and diesel vehicles are popular. It still is a country best seen with a rental car as public buses are just too difficult to deal with. Sometimes we just have to suck it up and pay the price.
And I find that driving in Ireland can be pretty tough, with narrow and crooked roads off the beaten path. It is just not a place for a driver with marginal driving skills.

You should definitely price it all out, but, I haven't run into one way drop charges when I am returning in the same country. I rented from Europcar and picked up in Colmar and returned in Tours, France. When I rented from Autoeurope, I picked up in Chartres and returned in Vernon, France. I didn't pay a one way charge on either of these rentals.

"You should definitely price it all out, but, I haven't run into one way drop charges when I am returning in the same country"

Yes, but Belfast (United Kingdom) and Shannon (Ireland) are in two different countries.

If you can re-jigger your trip to pick up and drop off the car in the same country, it will probably save you a lot of money. Or, you may decide that split among four people, the extra cost of a two-country rental is worth it.

Size does matter. My wife and I took our son and daughter in law to Ireland last year - four roller bags. We rented a BMW station wagon from Hertz that was a pleasure to drive and handled (barely) all our luggage. Since I used an American Express card to rent the vehicle, my insurance coverage was included in my Amex annual membership fee. I would check with your credit card company to see if that coverage is not already offered by them.