Good Morning! Get up early on Saturdays to avoid the crowd … My personal recommendation for a perfect Saturday start is … Borough Market http://boroughmarket.org.uk/ and the Monmouth Café https://www.monmouthcoffee.co.uk/. You can stroll along that amazing food market and get some bites for your breakfast. Also the Monmouth coffee does offer really delicious pastries and the best coffee on the river!
You could the follow the Thames Path passing lots of really nice places and exploring them starting with the Golden Hinde und The Clink Prison Museum. The next attraction would be the Shakespeare Globe Museum (which I can really recommend if you are not planning to see a play) and then the Millenium Bridge which you could also cross and visit St Pauls Cathedral and come back then later to see an exhibition at the Tate Modern.
There is usually lots going on on that Path … street musicians, some street artists … enjoy the walk! https://tfl.gov.uk/modes/walking/thames-path …. Next stop would be the Oxo Tower and Gabriel’s Wharf before the South Bank Centre and the Royal Festival Hall. Behind the South Bank Centre you can find a really nice and cosy streetfood market. So if you are hungry and want some rest, that’s the perfect stop. Watch also out if there is some new street art behind the buildings …
Finally you are reaching the famous London Eye. A ride is really nice and something I can recommend. And if you want you could cross the bridge and explore Big Ben and Westminster Abbey before you take the Thames clipper to Tower Bridge and the Tower …. and walk over the Tower Bridge back to the other side again. Shad Thames is a really nice area for dinner or my secret tip is Bermondsey Street with lots of nice restaurants and pubs. One I definitely recommend ist Village East to close off the day. https://villageeast.co.uk/