The Tube will be busy (standing room only unless you get on near the ends of the routes) but it will get you places in reasonable time. Buses and especially taxis will be very slow; I do not recommend taking a cab in the morning. In particular I do not recommend your idea of taxi from Paddington to Euston; take the tube instead.

There is little near Euston to see except the British Library if you're a bibliophile. Better follow stut's advice and take the tube from Paddington towards Westminster.

The Tower isn't small; it will take a couple of hours to go around the armouries and the Crown Jewels if you want to go there.

So, if you want to go to the Tower and have a quick look at other things, I suggest you leave bags at Paddington. Then take the Tube Circle line from Paddington (which is air conditioned these days, unlike the deep Tube lines, and has larger train carriages too) to Tower Hill. The Tower opens at 09:00 on Fridays. Then, take the Tube back towards Westminster, and look at the things stut mentions above. Proceed from Westminster or Embankment Tube to Paddington, retrieve your bags, the go to Marylebone to take your train to Brum.