Heathrow to Vauxhall by tube = Picadilly to Green Park then Victoria to Vauxhall.
When transferring take the route to the exit then when in the main ticket hall (but without going through the barrier) take the escalator down to the Victoria line - much easier than the 'official' route if you have luggage.
For a more exotic route take a bus (or taxi) to Feltham station then train to Waterloo (final destination) and get off at Vauxhall.