The best city in Ireland to visit is NOT Dublin, because it is far more English in feel than any other city in Ireland, and even the locals in Dublin say this. However, it is well worth a visit if you have a couple of days available. Just don't use a car in the city itself, it's much easier to get around on public transport.
Ireland doesn't really do cities, it's more rural than urban. You could visit any of the other cities though: Galway has a lovely 'on-the-edge-of-the-world' feel to it, and it the one city where you are like to hear Irish spoken by the locals. Still called The Town of the Tribes .... Limerick is smaller, greyer, but what most people don't know it that Limerick has a thriving arts scene, with poetry readings, performance nights, exhibition openings, theatre, music, etc. Great night life also. Cork is great too, I'm not so familiar with the nightlife there, but the cafes and pubs are excellent, and the arts scene is also great. Waterford is small and friendly, and if you get over next August for the August Bank Holiday Weekend, you shouldn't miss the Spraoi festival, it's great. Kilkenny has the Arts Festival and the Cat Laughs Comedy Festival, plus it's still got a mediaeval feel to it, and great night life.
I would suggest the same as many other answerers, hire a car and drive around the countryside. In October, you should be able to just wander and not need to pre-book accommodation, unless there is a local festival on somewhere. Check that the main tourist sites like Newgrange etc are open at that time of year before traipsing all the way out there, but you can also ask in smaller local places. The best places are not really on the tourist map.
The people are usually friendly. But don't get offended if they mistake you for an American. Not everyone knows the difference, so please allow for this very human error!
Ireland has become very expensive, in particular Dublin. A two-course lunch in a nice cafe can easily set you back 25 Euro per person, without wine. Pubs often do food at the same price range, but restaurants do tend to be a little bit more. What with all the other nationalities coming in to live and work in Ireland, the catering trade in general has undergone a radical transformation - you get a lot of Mediterranean and Asian influence in food in cafes now, it's actually quite rare to get colcannon, boxty, crubeens, bacon and cabbage and corned beef now. Potatoes are still a staple, but pasta, rice, and grains are also part of many menus. Don't be surprised if the waiting staff are not Irish, many places employ Eastern Europeans nowadays. Generally the quality of cafes etc. has improved a lot. If you're in Limerick city, eat at the Sage Cafe on Catherine Street. Their chocolate cake and their carrot cake are both to die for.