London is my favourite city very worth a visit. If you don't like tubes (tbh it can be quicker sometimes just to walk. Or, albeit slower than the tube the bus for ease of use jus pavement to bus no tube escalators)
'When a man is tired of London, he is tired of life' - Dr Samuel Johnson
Recommend:
Covent Garden The opera singer in the centre of the market if you can't afford The Royal Opera House. Also theatreland - stand in the 'Returns' queue and await tickets that have been refunded because a booking couldn't arrive - you must pay cash but I saw The Producers for £12 in a £60 Dress Circle seat at The Theatre Royal Drury Lane.
www.coventgarden.london/
The London Dungeons on The Southbank but then I'm a bit gory and always was as a child
www.thedungeons.com/london/en/
The Meridian Line at The Royal Observatory in Greenwich* so pleased my dad took us when he could when we were little and we got our certificate to say we had crossed the Meridian Line! It was a cheap day out and a moment in history for us. 
www.royalparks.org.uk/parks/greenwich-park/things-to-see-and-do/the-meridian-line
The Troubador Club - best in the evening one of my mum's old haunts from the 60s when she first left Scotland and lived in London. She took me a little while ago. Very small but great vibe and food. Best in the evening. Live Music! www.troubadourlondon.com/ I think you get off at Earl's Court.
Fortnum and Masons - The Queen's Grocers I lovvvve this shop and you can have afternoon tea at St James Restaurant with piano or string quartet accompaniment. www.fortnumandmason.com/?channel=ppc&gclid=Cj0KCQiAkfriBRD1ARIsAASKsQL9JlZotPZQj2nPt7YgvzZ60eYfMzf4jNFEFuJnCjeji5J3h4RPUZsaArcxEALw_wcB
The Natural History Museum and The Science Museum dear departed dad used to take ius there when he had no money and it was our weekend but we always found something interesting.
Leicester Square: Browns restaurant central but also reasonable price and variety for all tastes and budgets.
China Town - especially this time of year, Wonderful decorations on a big scale and fab atmosphere.
Southbank Second Hand Book Fair great buzz here and it's free as is The Tate Modern unless you want to see an Exhibition then you must pay.
Vidal Sassoon Hair Academy at Buckingham Gate (book in advance but Cut and Blow for £13 if you give up 3hours of life to a supervised trainee). We go every 3months love it and the building. It's number 58.
www.sassoon-academy.com/en/academy/uk/Models-Wanted
The Serpentine: Where my little brother thought he'd have a swim and risk Weil's disease (although Mum think's he probably infected The Serpentine 
The Courtald Gallery at Somerset House - even if you're not big on art they have a lovely outside area with fountains to have a quiet coffee and it's still central (I think Aldwych is the tube)
ttnotes.com/the-courtauld-gallery.html#gal_post_30216_the-courtauld-gallery-london-5.jpg
Hamleys www.hamleys.com/ I recall being told by mum as a child 'please don't touch the toys' but then mum being corrected by the shop assistant 'no, we actually encourage the children to play with the toys'. It's a cheery charming treat even as an adult.
We also loved seeing The Poppies at The Tower of London in 2014 but they have been moved since but keep an eye out for things like this. It was absolutely breathtaking. We couldn't afford the tickets but we made a good donation and stood outside and still saw all of the poppies and the Weeping Window.
Ronnie Scotts's Jazz Club www.ronniescotts.co.uk/#
Smithfield market: get in whilst it's still there! The best meat is at 2-3am and you don't have the Congestion Charge
The Royal Academy of Art opposite Fortnum and Masons in Piccadilly.
The Ritz - Tea in The Palm Court. You need to book but it's nice o say you have been. We went for a very special birthday and it was a treat just to say we'd done it.
Get yourself a copy of Time Out and make some plans. You won't know until you go 
Have also loved York, Truro, Edinburgh, Cheltenham, Stirling, Exeter, Barcelona and Paris
Wishlist - Madrid, Seville, New York, New Orleans, Athens, Rome and Milan