Ideas please - First time travelling to Ireland

I think the must do things will come down to your likes and dislikes. Maybe give us an idea of the things you like to do. The one thing to keep in mind is that even though it’s an island, smaller than your country, it still takes a lot longer to get around than you think. So, my best suggestion is to allot as much time as you can to Ireland, and narrow your focus so you aren’t trying to circle the whole island. Pick the northern area, or the western area, for example.
As for the family history, just try to have as much information as possible before you go, and try to verify it all in advance. Many years ago I spent an entire day in the records office in Belfast, hoping to find out more about my great grandfather, but I came away with nothing because I had little information to start with. I know more now because I have since found out his birthdate was wrong.

There are few records prior or 1800, and scant until 1850. Movements within the UK is like recording moving between states in the US. I have a ancestor that left "England" around 1750 and despite the websites and search engines, I've not moved beyond what my grandfather did in 1950.

Knowing your interests is pretty core to good advice. Giants Causeway is interesting, but only consumes a few hours.

With two weeks you could see quite a bit if you don't mind moving around a bit. Flying to Dublin and stay a few days to adjust then rent a car and travel west and then up into Northern Ireland returning to Dublin and then flying to Scotland from there. I seem to remember Rick Steve's in his Ireland tour book has a suggested two week itinerary as a good starting guide. As said above, don't underestimate the driving times especially on minor roads. You need to add about 25% more time than stated. What time of the year do you plan to go? We have always gone off season avoiding the summer months. Be sure to get an automatic car and a small one for country roads You will.a!so need the new permit to enter N. Ireland and Scotland. Two different currencies as well. They are both lovely countries. What do you want most, hiking, scenery, history? Those answers will help with suggestions.

As regards the family his whether you want to go to Belfast or Dublin depends on whether they lived in what is now Northern Ireland or the now Republic of Ireland.
It's not only Ireland where records can be scanty. I'm working on a relation who emigrated to New York, quite an important person. But I can't even trace his Ellis Island record, let alone stuff up state.

not to be rude but Scotland is part of the United Kingdom , the United Kingdom is : England Wales Scotland and Northern Ireland :United Kingdom of Great Britain (that is England Wales Scotland)
and Northern Ireland, you're British if you are from any of them four constituent countries of the UK

Ireland for reference is the constitutional name of the Republic, i always refer to the Island geographically as "the Island of Ireland", which has Ireland and Northern Ireland (Part of the UK ) on it hope that clears up any confusion :)

you will need a ETA to enter the UK so that will cover your trip to both Northern Ireland and Scotland
if you are looking family history in Northern Ireland it will be good to have a look at GRONI ( General Register of Northern Ireland)
https://www.nidirect.gov.uk/contacts/general-register-office-northern-ireland
remember you need Pounds for Northern Ireland and Scotland and Euro for ireland
ETA link : www.gov.uk/eta

I agree that you should find out as much as you can about your family and where they were from and plan your trip around that. You won't regret this, I promise.
You can schedule a consultation at EPIC (the excellent emigration museum in Dublin). https://epicchq.com/explore/irish-family-history-centre/
You'll see they want you to contact them well in advance if you're asking for research to be done. In the meantime, you can use a service like Ancestry. Depending on what time of year you're planning to travel and your comfort level with uncertainty, you could wait and see what they turn up at EPIC and then add those places to your itinerary. I wouldn't try this in summer, as accommodation can be tough to book on the fly.
Don't think twice about hiring a car since you're familiar with "wrong" side driving. You normally need one to get to the cemetery, which may be the only thing you can find.