You can also use contactless credit card to travel from Gatwick into London: just tap your card on the reader when you enter the platform at Gatwick, and again when you exit in London. This will cost £19.20 in the morning peak (0630-0930), and £10.70 at all other times. (If you get the Gatwick Express, to London Victoria Station, the fare is £24, but there is no need to do that - Victoria is not where you want to end up, and the "express" is little faster than the normal train.)

Regular travellers on this route know of a hack, which involves leaving the train at East Croydon, tapping out of the platform, tapping back in again, and continuing your journey to London on the next train. (The trains from East Croydon to London are very frequent.) This would reduce the peak fare to £13.20 and the off-peak fare to £8.80. But it's probably not what you'd want to do, having just arrived in the country with luggage.

You can also use contactless card to travel to Heathrow when you leave, whether you use Heathrow Express, Elizabeth Line or Underground.

And just to reiterate: as long as your credit card can be used as a contactless card ("tap to go", "tap and pay", "touch-free"), there is no need to get an Oyster card.