Galway or dingle

So many questions: Have you been to Ireland before? What time of year are you going? Will you have a car? What are your interests? What else do you have planned for your trip? If you are only planning Dingle and Galway, maybe fly in and out of Shannon instead? When choosing between two destinations, the best course of action is to research what is on offer in each and determine which appeals more to one's interests.

No I’ve not been to Ireland before. Yes we will have a car. And no direct flights to Shannon
We’re going at the end of october

With only 5 days at your disposal, and assuming that as a first time visitor a couple of those will be devoted to exploring Dublin, trying to see both places would be a logistical stretch. Better to pick one I think. My suggestion would be Galway since it's a much shorter drive from Dublin plus it's the gateway to the very pretty Connemara area ... a favorite for many of us here.
Keep in mind that you'll likely have to deal with some pretty unsettled weather and very short days that time of year. Sunrise in Galway is around 8:30 AM and it's getting dark by around 6:30 PM. Know too that Ireland observes daylight savings time the last Sunday in October, which will mean that it'll be getting dark around 5:30 PM. Short days indeed - and likely wet ones too.

With 5 days and it being your first visit, I would also recommend you stick to Dublin and Galway. In fact, you can save some headache (and money) by waiting to rent your car until you get to Galway (much easier.) There are buses that take 2.5hrs direct from DUB airport to Galway center (Citylink). Super easy after a long flight! Then you can head back to Dublin center to end your trip. Hope you enjoy your visit!

I don't understand the allure of Galway, TBH. For a first trip of 5 days in late October, I would either stay in Dublin (and make a day trip to someplace like the Giant's Causeway or even Howth to get a taste of the natural beauty and also day trip to Newgrange) OR head directly to someplace like Tralee (by bus from the airport), rent the car there and day trip to the iconic Cliffs of Moher, drive the Dingle Peninsula, stay in Dingle town or another small village, for an all "natural beauty" (and village) trip.

It is 2 1/2 hours each way from Tralee to the Cliffs of Moher, so that is 5 hours driving on what is a short day, sunlight wise, just to see the Cliffs.

I also don't understand this obsession with using the buses, rather than the trains in Ireland.

The train is every 2 hours, with a simple change at Mallow, and is as fast as the bus at 4 hours 10 minutes, and there is a direct bus from the airport to Heuston Station, Dublin every 20 minutes.

The bus takes 4 hours 5 minutes from the Red Cow Interchange in Dublin, you have to take another bus from the airport to reach the Red Cow, so it works out even in time terms.

I would also suggest just Dublin and Galway, in fact I might even suggest just Dublin. There are many places you can get to from Dublin either by train or guided day trip, and you wouldn’t have to rent a car and deal with driving and finding your way around. The train trip from Dublin to Belfast is only an hour, and Kilkenny would also be easy to do. Guided day trips from Dublin would get you to Glendalough, Newgrange and many others that will get you out into the countryside.