Greenwich

Use the river bus sponsored by Uber. No booking required.

This is the Uber boat by Thames Clippers website- https://www.thamesclippers.com/

Essentially boats run every 20 minutes from Battersea, Vauxhall, Westminster, the London Eye, the Tower of London and various other piers to Greenwich.
Your Oyster or contactless card can be used- but Oyster can't be topped up at Riverbus piers.

Are there boats with narrated tours? We thought to do this on the way over and then take the DLR back.

We were just in Greenwich yesterday. We took the tube. We arrived at 9:55. We didn’t book anything ahead of time and walked into each place. We also took a walking tour which we booked at the TI. There are two different tour groups at the TI. One takes you into the painted hall the other doesn’t. Ask first. We had a lovely day.
FYI, the cutty sark offers the opportunity to climb the rigging of the ship. We didn’t do it but watched 3 siblings, ages 65, 69 and 75. It looked like fun but I don’t know how adventurous you are, we aren’t very.

You can also get there on the DLR (Docklands Light Railway) from Bank Underground Station, by main line train various central London stations and by Bus 188 from Central London and Waterloo- the bus is a bit of a trundle through south London but is an option, which I did (not for the first time) this week.

We were planning on taking a boat over so that we could see the sights on the Thames River and then return on the DLR. Is there a boat you could recommend which would give us the tour of the sights on the Thames?

London City Cruises is one- they run from Westminster and Tower (of London) piers- https://www.cityexperiences.com/london/city-cruises/sightseeing-flexible-ticket/

and there's Thames River Sightseeing- https://www.thamesriversightseeing.com/

I've not personally used either since the Clippers started running, and that's a fair few years ago now.

Here a couple of websites that list your options:

https://freetoursbyfoot.com/thames-river-boat-cruises/

https://www.visitlondon.com/things-to-do/sightseeing/sightseeing-tours/river-tour

Scroll down to #11 on the above site and you will that there is an audio guide that you can download to use on the Uber riverboats.

If you want live narration, this is the only company I could find ( they offer various options):

https://www.cityexperiences.com/london/city-cruises/river-thames-sightseeing/

Edit: isn31c found a second company, so you have a choice of two for a narrated journey. But neither company guarantees live narration, from what Incan tell.

Last year we used Thames River Sightseeing (mentioned above) from Westminster to
Greenwich and took Docklands Light Railway back. We were staying near Waterloo station.

London Walks does various tours of the Thames and Greenwich. Lots of us like their tours a lot--perhaps they have one that suits your needs.
https://www.walks.com/

We did this one years ago. It might suit your needs.
https://www.walks.com/our-walks/greenwich-the-perfect-london-walk/

I also recommend the London Walks tour of Greenwich-including the boat ride down. It was a fair number of years ago, but we had a wonderful actor/tour guide called Nick who made the walk super-entertaining.

Unless you really want to get the boat down the Thames, the easiest way is to get the train from London Bridge or Charing Cross to Maze Hill station.