London 4/09-4/11

I'm confused. You're arriving April 9 (a Saturday), but have back-to-back theatre performances the Friday prior?
Could you clarify?

You say you are a "first time London visitor". You really should buy the Rick Steves' London guidebook. There is more information in it than we could ever give you on this forum, and you will need all sorts of info about transportation, hotels, etc., in addition to a list of sites to see. Rick also gives you a good map of London, which you will need. He gives you the location of museums, and opening and closing times. Also, go through previous pages on this forum. A similar question to yours has been asked about 30 times in the last month or so, with some really great answers and suggestions by forum members. My must-see list for your visit would be: Tower of London, St. Paul's Cathedral, British Museum, and Westminster Abbey. A stroll by the River Thames, Big Ben, Parliament, and Buckingham Palace would be on the list also. Possibly include the London Eye if you don't mind heights. And be sure to surprise your wife with a nice anniversary dinner someplace special. :)

(The OP messaged me privately and provided some details of his plans, which I'm sharing with his permission so we can all make suggestions. Arrival day is actually early Saturday, April 9, from Budapest. Check-in to hotel at Picadilly Circus. Double-dipping theatre performances that day: 2pm in the West End, and 7pm at the old Vic. Sunday is free, and departure from Heathrow on Monday at 6pm.)

My advice: Saturday will be a busy day. Between shows, you'll be able to eat somewhere quickly, but not much else.
Perhaps you can have a quick walk around one neighborhood before you head toward the West End. Assuming your first show gets out around 5:00pm, you'll have to grab a fast bite and then head to the Old Vic. I doubt there is really time for a sit-down meal in the theatre district itself. After eating, you can use the Underground to Waterloo and walk from there. If you're running late, catch a cab. Give yourself some leeway in your time planning in case you take the Tube in the wrong direction or get lost on the streets. (I've been to London 4 times and still got turned around headed to the Old Vic in 2009). Consider pre-ordering a snack or some coffee before the second show begins if available. You may want a caffeine or sugar fix by intermission of the second show. Others have already responded with the Big Hits of London. I always suggest seeing sites that are truly unique and quintessential to London. Tower of London is a must, and in my opinion, Westminster Abbey. If WWII is your interest, consider the Imperial War Museum or the Churchill War Rooms. You seem to like theatre, so the Globe is also an option, though one I'd skip in favor of the others mentioned. Walking by parliament /"Big Ben" can always be done in the evening after the other sites have closed. Not sure how soon you should be at Heathrow on Monday (3pm perhaps?), but assume you can only see one thing at the most.

You can't go wrong with an Original London Walks walking tour or one of their days out. Look at their web site www.Walks.com and you are sure to find walk to do morning, afternoon, nights (their pub walks are not crawls but more for people who want to know about the area they are walking in along with visiting a few historic pubs). They also do great days out where you take a train out to visit. there Old Hampstead Village Sunday morning walks is a good one.

Not much time on saturday except to eat and possibly walk between the two theaters. Sunday you are in a great location to walk Trafalgar Square, Westminster, Horse guards, Big Ben, Westmiinster Abbey (whatever time they will let you in between services), St James park, pass Buckingham Palace. Possibly a trip to the Tower and tower bridge. Walk back to Westminster on south bank via globe theater (they have tours). Sunday is a great day for mixing with the locals in Hyde Park, St James Park Monday- i would roam the top class shopping areas- Piccadilly, Bond St, Regent street. OR Westminster Abbey if you missed it Sunday

Planning on being actually at Heathrow at 3pm for a 6pm international flight is a good idea. Unless you take the Heathrow Express (15 minutes) or the Heathrow Connect (about 30 minutes), both from Paddington Station, figure on about an hour's travel time from the West End to Heathrow, whether by cab, hired car, or Tube. Heathrow is a huge airport and, frankly, much of it is an over-capacity mess. Play safe with the time.

Wanted to thank all readers for their comments/suggestions. We had a splendid visit to London 4/09-4/11, we did a lot of walking, sight-seeing and got in two theater dates to boot. BTW, not a fan of WarHorse.
Had a great time. One restaurant recommendation: Les Deux Salon near Trafalgar Sq. Excellent French food and not a wallet buster. Oh, and the French Room for a great Pub Experience.