Greenwich

Just wanted to add a few locations I really enjoyed at Greenwich. The Cutty Sark was surprisingly interesting. My Grandfather attended the Royal Hospital School there and it was special to walk the area and see the special photos in the Queen's House. Don't miss the Royal Naval College.

This blog post may help you determine where to get your picture taken
https://www.ottsworld.com/blogs/how-to-photograph-the-prime-meridian-line-for-free/

We walked under the Thames River (Greenwich Foot Tunnel) to the DLR Island Gardens station.

Pay for the museum. It's worth it.

We walked to Greenwich from our rental flat near Tower Bridge. We had a book on the Thames Path and historic sites on the way. . When we found and crossed the Prime Meridian, we took the photo. Then we caught the Thames Clipper, not a tour boat, back.

It seems we can take a DLR or Southeastern alone or with tube.

I am assuming you will have an Oyster card, in which case it doesn't matter if you take the tube, DLR or train, all the same.
Oyster is valid on all trains in London, no need to get an extra ticket for Southeastern.
There is also bus as an option. I did that once, takes forever - don't.

I would recommend the DLR because I like the elevated section, the contrast between the new buildings in the former docks at Canary wharf, and the old east end housing surrounding them.
The DLR is driverless, if you are lucky you can get the front seat and see out the front window.

The trains from Greenwich used to go all the way into Charing Cross which would have been very convenient for you, but now this would normally require a change at London Bridge.

I agree that it's worth paying for the Royal Observatory. If you get a combo ticket with the Cutty Sark, and if you buy that online in advance, you get a big discount.

The trick I learned is that the morning of your visit is far enough "in advance." So, if you're even thinking you may want to do this, go to the website now and create an account (MUCH easier on a computer than on a phone!). Then, on the day you go to Greenwich, you can just buy the ticket on your phone. I printed out the tickets at my hotel, but once I got there, I learned they could have scanned them off my phone.

Ticket website: https://www.rmg.co.uk/plan-your-visit/tickets-prices

If you don't pay for the Observatory, there is a small area where you can stand on the Prime Meridian without paying. Since the PM hasn't moved, the "old" directions will still work - unless they've closed off that area since I was there in September 2018. I see the prices have gone up since my visit, so anything's possible.

There is a Prime Meridian sign quite close to where I live in Waltham Forest. Take the Victoria Line tube or London Overground to Walthamstow Central Station. Get a 275 bus towards Barkingside. Get off at the Wadham Road stop. Walk back towards the bridge over the freeway that the bus just passed over and there is a paving stone set into the sidewalk on the same side of the road with the meridian line marked on it.