In London don't miss the British Museum. During their colonial days the British scooped up thousands of ancient artifacts from all over the world from every country they colonized, raped and pillaged. On the brighter side a lot of what they took has been saved from wartime destruction in those countries. London is awesome. Never been to Ireland. Enjoy your trip.

[killarney.ie]

You can spend a month in London and only scratch the surface. I go there nearly every year. As another poster asked, "What are you interested in?" Must sees for most people: Buckingham Palace (outside), Big Ben, The Eye (don't miss it -- best view of the city), Houses of Parliament, Westminster Abbey. All are close together and can be visited in an afternoon. Tower of London to see the crown jewels is a must. Diana's dresses are on display at Kensington Palace. If you l like art, don't miss the Tate Gallery. Some of my faves: Portobello Road, Nottinghill, and Camden Market. Hope this helps.

Try to find a Butlers shop for hot chocolate and chocolate. Enjoy being ripped off in Eire, I mean that in a good way, so be prepared to haggle. Head north to Limerick, there is a lovely castle/living museum there.
As for London, what part are you staying in? You'll need an Oyster card to travel as all of London Transport is cash free.
Eire is metric and you'll need Euros, so Kilometres, Kilos, Grammes, Litres, Celsius but also pints (20oz British ones, not puny 16oz) Ireland has better tea. Craic (krak) is fun, not drugs. Part of the EU. Generally public transport is good in Eire.
England/UK semi-metric - pints and miles. Possibly part of the EU, or maybe not. Plastic £5 & £10 notes. Northern Irish £5 notes are not legal tender on the UK mainland if you end up with some. Beware there are five different types.
Both places - make fun of Trump, we all hate him, so you'll get a sympathy vote. We drive on the left, remember that when crossing roads, however, if you go to Dublin, it's pot luck what side of the road they drive on there. Trams are fun in Dublin. Ask how to use them if you are there.

Sounds fantastic! I've a good friend traveling through Ireland as I type. I'll see if he has any tips. I've been a few times, but am not well versed in County Kerry there in southwestern Ireland. As for London... wow... that's a big question. Depends on your interests and how long you'll be there?