and also stop by the Imperial War Museum on the south shore of the Thames...fun to walk around Covent Garden area as well.
Heading up to Scotland, try to play some golf at either St. Andrews or Carnoustie, but on the way, stop in Edinburgh, as mentioned by others...and call ahead for a dinner reservation at "The Witchery"...just below the Castle itself...perhaps the best dining experience my wife and I have ever had...the service was impeccable as well. Also, virtually next door to The Witchery is the "Scotch Whiskey Experience"...if you're so inclined; can taste the differences from all regions of Scotland (e.g. the 'smokiness' of the whiskey from Skye)
I'd also recommend driving through Stirling and then on to Glencoe (sight of the McDonald's Massacre...very historic for Scots)...as you reach the road's summit before getting to Glencoe, there's a good chance you'll see and hear a Highlands Bagpiper by the side of the road...there's a pull out where you can stop and listen to him (maybe even give him a tip). Ft. William lies beyond...your choice as to how far - and wide - you then choose to go in the Highlands.
Now, as for Ireland...where not to go??...I'll have to bulletize my recommendations...(Note: it only takes <4 hours to cross the island on major roads)
Dublin...
>The Post Office
>Howth
>Trinity College...Book of Kells
>National Museum
>Temple Bar...esp. the "Palace Bar", which still has local musicians playing traditional Irish music.
>Guiness Warehouse...take the tour...get your Certificate for pouring Guinness...learn how to play the 'Bodhran'
>Jameson Distillery...another tour
>Grafton Street (walking...no cars)...then on to St. Stephen's Green
>Lots of places to stay, but recommend near Merrion Square or #10 Leeson St. Upper
>Ask locals where to go...
South of Dublin...
>Glendalough...St. Kevin's hermitage
Heading North...stop at Tara...then Newgrange...both historical "Musts"
Northern Ireland...
Belfast...
>Titanic Museum
>Downtown...walking
Head North...
>Giants Causeway
>Bushmills...Eat or Stay at Bushmills Inn
>Dunluce Castle
>Portrush...play golf at Royal Portrush...walk around the 'Raymore St.' area...good restaurants/views
Head SW to Sligo...play golf at Enniscrone...a hidden gem.
Stop in Galway...walk around town.
Stop at Lahinch...play golf...my favorite in all of Ireland/Scotland
Keep going south along the coast..
>Tralee...more golf
>Dingle...a must for Gaelic History
>Killarney
>Glengariff/Bantry Bay
>Blarney Castle...you don't have to kiss the stone...but the castle is interesting
On the south coast...
>Kinsale...a must stop...esp. "Charles Fort"
>Cork (Cobh)...many places to see, incl. University College, Cork
That's it for now...I think you'll need a few more trips to cover Donegal in the Northwest and Waterford/Wexford in the Southeast.
Hope this helps