Group your days by cluster, not by attraction list
London's biggest mistake for a three-day visitor is treating a bucket list as a route: crossing the city twice in one day for two "must-see" sites burns time and fare money that a slightly reordered plan would save. The clusters below sit close enough to walk between most stops, using the Underground only to jump between clusters, which keeps each day inside the flat GBP 8.10 pay-as-you-go daily cap for Zones 1-2 on contactless or Oyster, confirmed on TfL's own visitor fare guide. Walking also beats the Tube for anything under about 1.5 km in central London once you count the time spent going down to a platform.
Day one: Westminster and the South Bank
Start at Westminster, where Big Ben, the Houses of Parliament, and Westminster Abbey sit within a five-minute walk of each other. Westminster Abbey charges admission separately from the government buildings around it, so check its own current pricing before you go rather than assuming it is free like the national museums nearby. From Westminster, walk across Westminster Bridge to the South Bank — a flat, riverside 15-20 minute stroll past the London Eye toward the Tate Modern and Shakespeare's Globe, with no Tube fare needed for the whole crossing.
Finish the day back across the river near Borough Market for food, then take a short Tube hop back to your accommodation if it sits outside walking range — one journey easily fits inside the day's fare cap alongside the earlier crossing.
Day two: the free museums of Bloomsbury and Trafalgar Square
This is the day to lean on London's free national collections. The British Museum in Bloomsbury has free general admission and is open daily 10:00-17:00 (20:30 on Fridays); walk-up entry works, though the museum recommends booking a free ticket online for priority entry when it is busy. From there it is a walkable 20-25 minutes south to Trafalgar Square, where the National Gallery is also free general admission, open daily 10:00-18:00 (21:00 Fridays), with the same walk-up-or-book-ahead pattern.
Neither museum needs a paid ticket, so the only transport cost on this day is whatever it takes to reach Bloomsbury in the morning and return in the evening — a single Underground journey each way, well inside the daily cap. Covent Garden sits a short walk from Trafalgar Square if you want a third stop without adding a fare.
Day three: the Tower and the City, with one booking to make in advance
The Tower of London is the one big-ticket item in this itinerary that genuinely benefits from advance booking, and it is the most expensive single admission you will pay this trip. Historic Royal Palaces lists standard adult admission at GBP 37.00 (or GBP 40.70 including a 10% donation), and while same-day tickets are sold at the ticket office, the site itself advises booking online in advance "during busy times such as holidays" to avoid being turned away or facing a long queue.
After the Tower, Tower Bridge and the City of London's older streets (Leadenhall Market, St Paul's Cathedral) are all within a 15-25 minute walk along the river, so the rest of the day again needs no more than the single Tube or DLR journey it took to reach the Tower in the morning.
What this itinerary does not cover
This plan concentrates on Zones 1-2 and does not route you out to Greenwich, Kew, or Hampton Court, all of which sit further out and would each need a separate fare beyond this cap-friendly core. It also does not include evening West End theatre or a river cruise, both of which are worth adding as a paid extra on top of, not instead of, the walking plan above — build them into whichever day has the lightest daytime schedule.