Unsure how to plan my 6 nights in Ireland

It's at least a 3.5 hour drive one way from Dublin to the Cliffs of Moher. There's about 11 hours of daylight in Ireland in October. So driving to the Cliffs for a day is doable. You could rent a car just for the day to do so. Or you could take an organized coach tour. I've never taken one but a friend did. He said it was a very long day. If Galway is of interest you could catch the bus to Galway upon arrival in Dublin. It picks you up outside the terminal. That would maximize your time and minimize backtracking. Belfast for a day is certainly doable as a day trip from Dublin via bus or train.

IMHO, Moher is over-rated. On the other hand, I expected to be underwhelmed by the Titanic Experience and found it impressive. Assuming you've never been, with just five days, I'd do Dublin (with a day trip to Newgrange/Boyne) and Belfast (with a day trip out to Giant's Causeway etc.). Your first night will be to recover from jetlag, so don't expect to do much on your arrival day. Both cities have lots of nightlife including trad bars with craig and sessions and packed madhouses in Temple Bar singing Galway Girl or rock'in roll, if that's your preference. There's no shortage of day stuff as well. Arrival/Departure times will impact how you sequence that.

Our first trip to Ireland was six nights (without jetlag issues as we were coming from the UK) and our nights were Brae, Killiane Castle, Killorgin, Doolin, Galway, Dublin and we loved what we saw/did/hiked/pub'ed, but I was exhausted from the driving and I wouldn't recommend anyone repeat our trip. That first trip was in 1997 before the internet changed travel, and you stayed where you got tired of driving. We've been back three times (8 weeks total) since to fill in the gaps and have next year's 3 week trip planned (and the 3wk trip after that). Ireland is endless.

I'm another one who found the Cliffs of Moher much over rated. You really need to see them from below on a boat trip. They are also completley the wrong way too...

I do like jjgurley suggestions of Dublin (with a day trip to Newgrange/Boyne) and Belfast (with a day trip out to Giant's Causeway etc.)

Well, I loved the Cliffs of Moher, and found it a great experience to stand at the top in the wind, and look down the coast. It’s quite dramatic. But, with so few days the Cliffs are a long day trip from Dublin. I did a guided trip from Galway so it was much better. I would look at some of the day trips from Dublin and see what other stops they make, and if you don’t mind the long day then go for it. Although, October may be late in the season and they may not be running by then. I booked all of my day trips in advance, too, because the popular ones fill up, so waiting to see what the weather is like may not work. I would also suggest just staying in Dublin because each time you pack up and move you will lose time to touring. Belfast can easily be done as a day trip from Dublin, and it’s an easy city to get around by transit. You could do a black cab tour as well as the Titanic museum. Six nights only gives you five days, maybe less depending on when you have to be at the airport to fly home. You could get in a couple of guided day trips, go to Belfast for one day, then spend the rest of the time just seeing Dublin.

Agree with those who found the Cliffs of Moher overrated; we thought it was a sort of Disneyworld of Ireland. The sea cliffs at Slieve League on Donegal Bay are far more impressive, IMO, and you can even book a view from below on a small fishing boat, weather permitting. You might consider the idea of a Dublin-Belfast-Donegal triangle, to be completed in whichever direction suits you.