Been to London 7 times and here are some personal tourist ”highlights” from the top of my head:
Berwick Street – take a walk and pretend that you’re on the cover of THAT Oasis album.
Imperial War Museum – impressive collection if you into that kind of stuff.
National Gallery – no entrance fee, so just waste 5 minutes to take a look at Van Gogh’s sunflowers and leave again (or stay longer if you like) 
A boat trip to Greenwich if the weather is fine is always a good idea. The area around the observatory is quite lovely.
Notting Hill is nice, but certainly didn’t remind me of the movie at all.
The Fuller Brewery tour was great (if you like beers that is). The tasting afterwards left no one thirsty. 
Stay away from Little Venice and Oxford Street.
Frank,
I haven't lived in Dublin for about 6 years but am moving back there this June (woohoo). I would suggest getting in touch with Niall (http://www.nialler9.com/) and chat with him about music to check out. He has a good feel for live shows.
good place to eat: http://www.ukiyobar.com/ A mate of mine runs it so if you go in ask for Duncan and say you know Colin from cubikmusik. :)
Also good: Cafe Irie in Temple Bar- good soup/ sandwiches and despite the somewhat cheesy name it a very nice place to hang out.
Temple Bar, and surrounding areas are good for shops (stores). Off Georges St. is http://bigbrotherrecords.com/about.html.
Good bars: Old school-Mulligans on Poolbeg street (close to Trinity College in the city center) has the best Guinnees in Ireland (and perhaps even the world?). No cell phones allowed! :)
Kehoe's on South Anne Street-this is off the main pedestrain street (Grafton Street). Also check out, if into literature, http://www.dublinpubcrawl.com/
I agree with Harry BombShelter re: Guinness tour. The sky bar is an amazing place to drink a pint.
Check out these 'short and sweet' guides. Not sure how up to date it all is: http://www.thesugarclub.com/shortandsweet/.
email me if you wanna know anything else mate. I hope you have a great time. Cheers.
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