In message , midunjen
writes

We are taking the underground from Heathrow Airport with the ending
destination of Chester. From looking at a map, we will need to take
the Heathrow Underground (from terminal 3) to Euston station, then
catch a train from Euston to Chester.

Correct.

My question is: which Euston station is a National Rail station? There
are three on the Tube's website.(Charing Cross, Square, City Branch).

Just plain "Euston" is on the both branches of the Northern line and on
the Victoria line, and will take you directly into the National Rail
station.

Euston Square is less convenient if you have luggage as it is several
minutes' walk away.

I was hoping there was a direct train but it doesn't look like there
is.... any suggestions?

The most direct tube route is Piccadilly line from Heathrow and change
to the Victoria line at Green Park. The interchange there is all
underground but it is not ideal if you are carrying a lot of luggage. An
alternative would be to transfer to the Northern line at Leicester
Square - but the Northern can get very crowded at some times of day!

If you have ample transfer time it may be worth considering taking the
airbus which runs directly from all four terminals at Heathrow to
Euston, as this will involve no change. There are two buses an hour and
it would be wise to allow 90 minutes for the journey as London roads are
rather congested!

--
Paul Terry