From Cork to Belfast is pretty much a southwest to northeast trip. I don't know the direct inland routes, but I have taken the western route. It could take you through Galway via Ennis which is great for music. From Ennis it's only a short detour to the Burren, Cliffs of Moher, and Aran Islands.

Further north I recommend Sligo town and the surrounding "Yeats country" - Ben Bulben, Glencar lake etc - and on to Donegal town. Lots of storied music area north of Donegal such as Teelin and Gweedore; lovely scenery at Glen Gesh Pass and Glenveagh National Park. The North coast is great as well (Bushmills, Giant's Causeway).

Or you could go straight East from Sligo through Letterkenny and Co. Fermanagh and Monaghan, which I haven't seen but am told are lovely.

Any way you go, I'm sure you'll have a great time.