Victoria Station to Hampton Court Palace

That is a good question: I found that it does not seem possible to go without at least one or two changes. Cheap method is tube/bus, and more expensive (maybe a LOT more, check) is the train (always cheaper if you book 90 days in advance).

https://www.google.com/maps/dir/Victoria,+Victoria+St,+London+SW1E+5ND,+United+Kingdom/Hampton+Court+Palace,+Hampton+Ct+Way,+Molesey,+East+Molesey+KT8+9AU,+United+Kingdom/@51.4377832,-0.3240172,12z/data=!3m1!4b1!4m14!4m13!1m5!1m1!1s0x48760521aaebdc4d:0x5d364f36086cfd15!2m2!1d-0.1438979!2d51.4952103!1m5!1m1!1s0x48760b0e0e308247:0xe51344c894b5a35!2m2!1d-0.3377623!2d51.4036128!3e3?entry=ttu

On inner suburban routes such as this there are no advance fares- it doesn't matter whether you book 9 minutes, 9 days or 90 days beforehand. The fare is always the same, in the London zonal system.
There are multiple ways to do the journey- District Line to Richmond, then South Western Trains, or bus to Waterloo, then South Western trains being the most obvious.

There is an option all on the train, travelling from Victoria to Clapham Junction changing there for Hampton Court Station. As a single mode journey it is a bit cheaper. There is though a bit of a walk though at the end. Train to Kingston then bus gets you much closer.

We went to Vauxhall on the tube and then took the train to Hampton Court. Very simple!

If you go by train you will one way or another wind up on a South Western Railway (SWR) train which originates at Waterloo Station and terminates just before it tips into the river Thames at Hampton Court Station. The Palace is a few years to the river, over the bridge and then an immediate right turn into Hampton Court Palace.

As in most things to do with transport in London there are many ways to achieve the goal, all with pluses and minuses, and all in their own way valid. All the ones in previous answers to this thread are valid, some more or less easy, and some more or less convenient.

All the main stations in London serve different areas of the city and surrounding parts of the country.

From Victoria the Southern and Gatwick Express trains all cross the river and then primarily serve destinations south of the city, in Surrey, East Sussex and West Sussex, all the way down to Brighton and east and west from there. There are also some Southeastern trains which make their way through Surrey into Kent.

None of the trains from Victoria go west towards Hampton Court.

Just across the river from Victoria, and off to the side a bit, is Waterloo station, the busiest in the country, where the SWR trains fan out to south and west and southwest of London, reaching all the way down to Portsmouth and parts down that way, including the trains to Hampton Court.

There is a station south of London called Clapham Junction. This is the busiest interchange station in the country. Handily many Southern trains from Victoria and SWR trains from Waterloo stop there, and it is relatively straightforward to make connections between the two. So that is how the Victoria > Clapham Junction > Hampton Court train journey can work.

The first station outside Waterloo heading toward Hampton Court on the SWR train is Vauxhall, which is also served by the Victoria tube line, so the person mentioning that can catch a bus to Vauxhall or the Victoria line tube to Vauxhall (whichever is more convenient to where they start) and catch the train to Hampton Court from there.

and there are many other options but I will leave those for now...

Hope this helps...

and I agree that there is no advantage on these routes to Hampton Court to buy the ticket in advance, unless you want to. Same price.

More detailed fare information-
Pay by Oyster or contactless Off Peak fare each way from either Victoria or Waterloo main line stations is £4.80 each way= £9.60 return (£7.70 Peak)
Buy a physical ticket and the fare is £8.60 Single at all times or the return is a £15.20 Off Peak Z1 to 6 Travelcard.

Travel from any Zone 1 tube station via Richmond- Oyster/Contactless is £6.40 each way Off Peak., £9.40 peak

Travel from any Zone 1 tube station via Vauxhall or Battersea Power Station (LU)/Battersea Park (SWT)- out of station interchange or Battersea Power Station (LU)/Queenstown Road (SWT)-out of station interchange or via Elephant and Castle- Oyster/Contactless is £6.20 each way Off Peak or £7.90 Peak

Zone 1 to 6 has a £14.90 off peak Daily Cap (Hampton Court is in Zone 6)

Kingston is also in Zone 6, then add on a £1.75 bus fare= £6.55 each way off peak, £10.45 peak

You can also get a bus R68 from Twickenham to Hampton Court (London Terminal Stations to Twickenham by Oyster etc is £4 Off Peak or £6 Peak) or bus 111 from Heathrow CBS.
From Kingston the buses are numbers 111 and 411

That seems to be all the possible fare combinations.

The net answer is that the most expensive method (not the cheapest) is to buy in advance.

The total cheapest way would be to use the London bus cap of £5.25 and use bus 27 Central London to Hammersmith, 267 Hammersmith to Twickenham, R68 Twickenham to Hampton Court- journey time 2 hours each way from Euston, 1hr 45 from Paddington.

From 3 March the anytime Zone 1 to 6 fare cap rises to £15.60 and the Off Peak Travelcard to £15.90. The bus cap is unchanged.

Well, golly gee, I was wrong about advance fares. I should have known not to generalise anything to do with the English train system. I was wrongly assuming that what usually applies to intercity train fares, would apply to this case, which it does not. I got nailed going from Richmond to Twickenham once, and another time to Weybridge, Surrey, so I try to pay more attention now. London is still my favourite city, regardless of its foibles, but seemingly little costs can add up.