We are going to Ireland for our first time (not including a 24 hour layover) for two weeks in July. Our goal is to try to get a taste of as much as we can; blending the big "must see" tourist destinations with the smaller, local activities, music, and pubs. Our plan so far is this: arrive in Dublin and spend two nights there before heading up to Belfast for three nights before heading to Westport for a night and then heading south on the west coast. On the way to Belfast we plan on stopping at the Slane castle and distillery and Newgrange so we won't be getting to Belfast until later in the afternoon. While in Belfast, we want to do Bushmills, Giant's Causeway, and the rope bridge and St. George's market. The question is, is it worth it spending a third night in Belfast or does that pretty much cover it? We have no interest in the Titanic exhibit and Westport looks absolutely stunning so we're thinking about swapping a night in Belfast for another one in Westport, but we don't want to feel like we're selling ourselves short in Belfast.
I assume you'll have a car. Otherwise, I'm not sure how you'd get from Belfast to Westport in a day. If the only thing you want to see in Belfast is St. George's Market (which I left after about 5 minutes, BTW, and I LOVE a market), why not stay in Bushmills instead? You can stagger back to your accommodation after the distillery tour, and it's a good access point for the GC.
There is no such thing as "must see" destinations. See what you want to see. And maybe pick out a couple of things to see on your very long drive from Bushmills/Belfast to Westport. There are loads of them. Derry, Donegal, the Slieve League, the Sperrins.
Is your plan for Northern Ireland to go to St George's market on the afternoon of day 1, then a day trip to the Causeway Coast on day 2, and drive to Westport on day 3? I would set aside a whole day for Causeway Coast (Bushmill's/Giant's Causeway/Carrick-a-Rede rope bridge)- there are other places you can stop as well like Dunluce Castle or Ballintoy Harbour- either in your own car or on one of the organised tours. There's lots more to do in Belfast particularly if you're interested in recent history, like a black taxi tour or Crumlin Road goal, and there's great places to walk and view the city from Cave Hill. There's also (I believe) the only surviving White Star line ship, the SS Nomadic, which was a tender for the Titanic (tho if you're not interested in the Titanic exhibition that may not interest you either). I've never been to Mayo so I can't say how it compares to Westport but I personally love Northern Ireland- it's all about what you are most interested in.