Just seen this Faringdon, good luck, should be a great day. Funnily enough, I had an NHist email today, talking about privilege guided tours to beat the crowds?!
I see that a cab fare, according to Mr Google and his new friend Arty Instein, will be some £15 to £20. Plus a tip of course?!
Possibly money is no object, though if you've paid a prime-time fare from Swindon or Didcot, your card(s) have already suffered a major blow?!
Padd'ton to Nat History is quite simple, by Tube (Circle - though it isn't a circle any more, most confusing?!) to South Ken works, but I always think that long long walk through a claustrophobic tunnel (often packed like sardines!!) is no fun?
You could instead go 2 stops to Notting Hill Gate, get out, walk east (towards Hyde Park), and find the bus stop for Bus 70 on the south side of the road. That bus comes down past Queensway, then turns left just before Nott Hill Gate station and drops down to the Musuem, stops right there. If you have a bus pass, that works in London too, but your GD has to be paid for (use your card). As folk have said, you can pay for the Tube by tapping your card (oldies do NOT have the bus pass equivalent, unless you live there), but with GD to pay for as well, you just need to co-ordinate that tapping. If in ANY doubt, just talk to the attendant, that's what they're there for, and usually delighted to help (especially with GD in tow?!).
Coming back, just reverse the journey, BUT, if by then you need a G&T (or 2?!) and a foot massage, maybe the black cab is the answer?!