Some great advice in this thread. Glasgow has wonderful free art galleries (Kelvingrove). Edinburgh is incredible, especially during the festivals but accommodation will be through the roof; apart from the culture on offer you can just walk and walk and soak up the glorious architecture (maybe walk the Water of Leith to Leith and get the bus back).

Fort William is an excellent base to explore the Highlands (Jacobite steam train, places Jock has mentioned), as is Inverness. Inverness is quite a lively small city and close to Loch Ness, Fort George, Culloden, Black Isle. Ullapool is about two hours from Inverness and also a stunning location with the route from there to Kylesku being a favourite but there are too many excellent places for one three week visit as said.

I think you are going to have the most marvelous trip.

I think midges are overestimated, unless you choose to wander up hills where there are lots of farm animals, where you are likely to be eaten alive in my experience (maybe more so in Ireland).