Train
There are two choices to travel towards London by train.
Heathrow Express
As the name suggests, this is a non-stop service from terminals 2, 3 and 5 (T4 passengers need to take a different train and change).
The service is quick (15 minutes usually) and starts early and finishes late. Frequency is mostly every 15 minutes.
It is, however expensive. A one way ticket booked in advance may be £17 or more. Ticket on the day can also be more. It is one of the UK’s most expensive train services. Oyster and contactless are not allowed.
The train goes to Paddington station. If you are staying in that area, it’s great, otherwise you will need further transport eg taxi, bus or tube.
The Elizabeth Line
This is a very extensive line that goes from Reading and Heathrow in the west, across central London to Abbey Wood and Shenfield in the east. It’s not really an underground service, but it does accept Oyster etc and is cheaper than the Express.
The Elizabeth Line starts at T4 and is the first service you may need if you want the Heathrow Express. It follows the same route to Paddington (but with many stops along the way, including a useful change at Hayes if you are heading west). From Paddington, it goes through central London (eg Bond Street, Tottenham Court Road and Liverpool Street station) then onwards towards Essex etc. There are interchanges along the way.
It gets very busy. The stations are huge but very deep so getting in or out with bags can be a task.