Heathrow Connect

Yes. https://www.heathrowconnect.com/station-information/heathrow-airport

Do keep in mind that if you happen to just miss one that you will be kicking your heels for nearly half an hour. How are you completing your journey?

The train will take you from Heathrow to Paddington, but then you need to get to Kensington (by Tube, bus, taxi?), which means the Tube from Heathrow to Kensington may not be a bad idea in the first place.

Also, Kensington is a bit vague, so if you say exactly where you're going then we can offer suggestions on the best options

We will be going to the Copthorne Tara on Scarsdale Place in the Kensington area. We thought we would take the Heathrow connect to Paddington where we could then buy 7 day paper travel cards from the rail station. Then we thought we would take a tube, uber, or taxi to the hotel. Do you think it would be best to just take a tube from Heathrow to south Kensington station? We thought since it was rumps hour, the tube would be packed.

Tharp, I've just looked up "Scarsdale Place, Kensington" The nearest tube station is High Street Kensington, NOT South Kensington, that is 2 stops away. Click here for map.

The trains at Heathrow are all well signposted, they have to be for people with jet lag.
If you haven't got a lot of luggage you could take the tube from Heathrow (with change) to High Street Kensington, or, if you take a train to Paddington, you can take a short tube ride from there to High Street Kensington.

If you have a lot of luggage and do decide to get a taxi from Paddington, tell the driver "Scarsdale Place, Kensington", NOT South Kensington.

We were planning on taking a taxi to our hotel in Chelsea. How do you go about pre-booking one? Alternatively, would we be wiser to take the Healthrow Express to central London and take a taxi from there? We are both in our upper 60's and don't want to wear outselves out climbing lots of steps out of the tube carrying our luggage. (We do plan on traveling as light as possible but still.....)

bherrington, a taxi from Heathrow to central London is painfully expensive, and the slowest of all the possible methods of travel (bar walking). In you case I would recommend getting the Heathrow Express or Heathrow Connect train to Paddington station, and get a taxi from the taxi rank there.

Tharp,
So we wished we hadn't taken the H-connect for these reasons. The H-express has a 2 people for 50 special, and it would have been completely worth it and wished we did. At Heathrow, we just missed the H-connect, and waited 30 minutes for the next one. During this time, three H-expresses came in and left. Our train arrived, hopped on, and then painfully slowly crawled to a stop at every single station (7) along the way. Guess how many more H-expresses we counted flying past us without a stop? yes, three more. finally, over an hour later, we were at our station. When we returned back to heathrow, guess what repeated itself? yes. So we lost over 2 hours of vacation time just to save 7.50 pounds total.

As Jean-Paul mentioned, the Heathrow Express can be close in price to the Heathrow Connect if you get the DuoSaver ticket. This only applies if there are two traveling together, and you are returning to Heathrow within 30 days. It is a roundtrip ticket for two costing £50, or £12.50 per person per trip. That is only £2.40 more than the Connect. And if your incoming flight arrives at T5, you will have to transfer to T3 ( using the Express) to catch the Connect.