how to get to Hampton Ct Palace

I would suggest Option 1. We were recently staying in Richmond and we took the bus to and from HCP and it was a long ride with traffic and a crowded bus requiring stops at most of the bus stops.

The boat takes one hour. POV of a tight fisted local- option 2 or bus all the way if I can work it out. £5.50, hopefully £3.50 depending on connections and the 62 minute rule.
POV of a tourist being guided by a local- Option 1 out, Option 4 on the way back, possibly vv for the wow factor.

I've taken Option 1.

From South Kensington it is the simplest option. Take it one way. (The train you pick up at Wimbledon is the same train that leaves from Waterloo.)

Regarding the District Line--make sure you get on the right one. It should be headed towards either Wimbledon or Richmond. The electronic board above the track at the South Kensington station will tell you where the approaching District Line train is going. It will also have it on the front of the train.

If the weather is nice, and you have the time, take the boat back.

If you pay via contactless, there are no ticket barriers at Hampton Court. Remember to tap in and out. The machines are on the side near the entrance to the tracks.

thank you everyone!! One question about taking boat back if weather permits -- I already plan to take Thames river cruise from Westminster to Greenwich - on a different day of course. Would this ride from Hampton Ct to Richmond a bit of a repetition?

If you use option 1, you can use your train tickets to get a 2FOR1 discount at the admission desk. You'll need to download a voucher from the Days Out Guide website and show that with your train tickets. This is only valid if you buy train tickets, not if you tap in and out, and not available in school holidays and at Christmas.
https://www.daysoutguide.co.uk/hampton-court-palace

I am actually thinking of the less expensive uber boat, mainly because of the frequency and convenience. And I don't really need a narrated experience, lol

Richmond to Hampton Court is what you think of as river, Westminster to Greenwich is broad tidal river where ocean going ships can still reach. Indeed if you are lucky you may see a cruise ship in the Pool of London, by HMS Belfast, and you are very likely to see freight barges.
These include the rubbish (trash) barges to Essex.
Being tidal the river is in a constant flux, very different at High tide vs Low Tide. From Tower Bridge onwards you are seeing the original London docks, before they moved down to Tilbury and now to Thames Gateway even further towards the sea.
At Deptford (the junction of the River Lee, navigable far into Hertfordshire) , on the North Bank, you also pass Trinity Buoy Wharf with London's only true lighthouse - used for testing and training until 1988, now open to visit at weekends.
If you visit the wharf by land there is an original American Diner (a true Diner) there, still used as a Diner.
If you are there in September it is worth while taking a sailing on Paddle Steamer Waverley from Tower Pier. Tower Bridge lifts for you and you can sail to Whitstable or Southend or sometimes further to Great Yarmouth or I did Folkestone once.
You see river evolving into Estuary and eventually sea.

If you stay on the Clipper to the terminus at Barking you transit the Thames Barrier and get to the newest part of London- brand new developments in Barking Reach. Then London Overground train from Barking Riverside back into London. Beyond where most tourists get to.
But there are the scant remains of Barking Abbey, used by William the Conqueror before the Tower of London was built and before Dissolution one of the most important Christian sites in England.

Another trip worth doing, when you can get it, is the Canal trip from Little Venice around North London and to the Thames at Limehouse Docks.