My plan for my tubewalk was to walk each Tube line, in the order in which they were opened, from end to end and including all sub-branches. If there was an obvious chronological direction to take then I followed that, otherwise I tended to walk from west to east and south to north, so the sun and prevailing winds were behind me.
Here's a summary of my schedule. I spread things out so I could spend time writing up my experiences in between, so you could squeeze the same walk into a much shorter period if you wanted, but I think it's good to take your time. There's a lot to see out there.