From Euston, I would get the tube to St Paul’s. (Northern line - black - to Tottenham Court Road, then Central line - red- to St Paul’s). You can use any debit card with a chip in it to pay by tapping it in the barrier. I would strongly recommend avoiding being on the tube between 8 and 9am, as it will be very busy and people will be fast and impatient. Fridays should be fine then though.
After you have seen St Paul’s (nice and always clean toilets are in the crypt) , go to the gardens outside, and be sure to see the reflection garden, where you can take an amazing picture of the cathedral dome reflected in the water.
Walk down to the river and cross using the Millennium bridge. You will see Tate Modern in front of you and pass the Globe. Walk along the river path until you get to Tower Bridge. You will pass Southwark Cathedral (which is worth a look), lots of cafes and Borough Market. Cross Tower Bridge and go to the Tower of London. To see everything takes several hours.
From here you can go down to the river again. This time on the side furthest from Tower Bridge, and get a Thames Clipper Uberboat from Tower Millennium Pier to London Eye. It will be RB1 or 2 during the week, RB6 at the weekend. Make sure you get on the westbound service. The staff are used to being asked for directions. No need to prebook this, it is first come first served. Again you can use a debit card.
Get off at the London Eye stop, and you will be right by the Eye queue.
After the Eye, walk to Waterloo station, and get on the northern line back to Euston.
The museums on top would be too much I am afraid