Split the difference. Achille Island is at the top of Mayo, just north of Westport, which I agree is beautiful but filled with lots of other people looking at its beauty. Achille is a bit desolate, far more sheep than people, dramatic and drivable without a ferry. If you're heading south from there, take the inland route through the Mweelrea mountains. When you hit the Galway border, get ready for the tour buses.
You can't lose in the northwest, just take your time and don't think you have to get someplace else. The beauty of the place is subtle at times and takes some relaxing to appreciate. A pint or two can aid this process.