Train from London to Hampton Court

Best as I can tell, weekdays there is a direct train every 30 minutes at 6 and 36 past the hour and it takes 37 minutes. Hampton Court is the end of that line, so that destination should be posted on the train board.

You can get a discount with a return ticket, or just pay with contactless credit card each way with no discount.

Take the Northern Line from Tottenham Ct to Waterloo.

Probably not the best choice but there is a scenic route if you take the district line to Richmond and then the bus to Hampton Court -- note that there are three stops, the main entry drive, the staff side entrance (easiest), and the maze/gardens. All called Hampton Court.

I would take the train from Waterloo.

Many of the sights in the UK have info on how to get there by public transit. Hampton Court’s info is here:
https://www.hrp.org.uk/hampton-court-palace/visit/getting-here/#gs.l7a1it

The reverse is the opposite of what Sam says. A physical ticket costs £16.60 return off peak or £23.40 peak, contactless or Oyster is £7 each way off peak (outbound after 9.30 am) or £10.40 peak.

bus to Hampton Court -- note that there are three stops, the main entry drive, the staff side entrance (easiest), and the maze/gardens.

note that while a portion of the gardens, the Wilderness, and the ancient Maze used to be open to the public and accessed for the bus stop indicated, that is no longer the case. All entry to all parts of the Hampton Court Palace and the gardens is only through the main entrance or the entrance to the Rose Garden, all admissions are charged.

On a nice weather day we recommend taking the water taxi back to London. The journey provides the experience of traveling on the Thames, going through several lockes and enjoying the riverside views. Perhaps most enjoyable is the experience of traveling at a pace commonly used for hundreds of years by people journeying to/from London. The taxis offer interior and exterior seating along with a bar offering a variety of beverages and snacks.

Getting to Hampton Court is easy. Just use your Oyster card or credit card on the tube to get to Waterloo Station, then get on the train from there to Hampton Court (again, you can use contactless payment for the train at Waterloo).

Once you get off the train, turn right, walk over the bridge and you'll see Hampton Court Palace,

To find the times (although they come pretty frequently), just go to https://tfl.gov.uk/plan-a-journey. I think the trains leave from Waterloo about every 30 minutes.

Hampton Court is within the London transport zones so you can use an Oyster or just tap on and off with a card. Train journeys within the zones are part of the same payment system as tubes and buses.