So we've gotten into Gatwick Airport, next you've got to think about how to get into the city. If you aren't smart about this, you'll literally end up paying more than you've paid to fly there.
The most obvious choice is the Gatwick Express. Straight into London Victoria in 30 minutes, no stops in between.
Now I will give you my advice on how we do this part, but I'm not 100% sure this is exactly for certain the cheapest way to do things. You read that booking online in advance is always the cheapest way, but we don't do that, so I may not be giving you the best advice here, but I'll share anyway.
You'll get a little shuttle bus from the plane to Gatwick Arrivals. Once you are in the building you'll pass through the baggage reclaim, then exit into an open area with restaurants & shops. As soon as you get into that large lounge, turn straight to your left, almost turning back on yourself and you'll see a train ticket desk. There's usually three desks open, and no real queue to speak of. You'll want to ask for a ticket that gets you into the city, and back tomorrow, but also works as your run around Underground ticket for both days.