Looking for the best way to Plymouth by train from Heathrow terminal 2.

I booked my ticket for London Paddington to Plymouth through Great Western Railway (for a March date). You can try plugging in your dates there (leaving from London Paddington) and see what available options there are. I purchased an Advance ticket which is quite inexpensive, but non-refundable. If you need a flexible ticket, it will cost you dearly. For example, I just plugged a date in mid-May for tickets. The Advance ticket was £41 and the Anytime ticket was £161.

So it might be beneficial for you to either book a non-refundable ticket for later in the date to make sure you get there. I would also get transport from Heathrow into London separately. That way you can allow for any delays and so on.

Firstly Trainline are OK for some research but you are much better booking with the train company that runs the trains. In this case GWR -www.gwr.com. Despite their advertising Trainline don't sell tickets any cheaper, will often charge a small admin fee and put a third party between you and the provider if anything goes wrong.
There are a variety of tickets you can buy. Roughly they are:-
Advance - the cheapest but you must travel on a specific train. Miss it and you need a new ticket. Prices rise as tickets are sold on specific trains.
Off-Peak - more expensive but can be bought on the day and you can travel on most trains outside rush hours
Anytime - most expensive, can be bought on the day and you can catch any train
Advance are obviously tricky if you are flying in as it only takes a delay on the flight to create a problem. Or you allow enough time and are left watching trains leaving that you can't get on.
Both Standard Class and First Class are available on these trains. First Class is nice if you can get a good price but Standard is perfectly adequate.
You should be able to manipulate the timings to give you more time to change at Paddington, although 16 minutes should be enough.
There are other options to catch the train. You can take a rail/air coach from Heathrow to Reading and pick up the train there, but I don't thing these days it saves much time or money. There's also a route on the Elizabeth Line to reading to that requires another change at Hayes & Harlington which you might want to avoid.

To follow up Mardee's post looking at Mid May you can also get a Super Off-Peak ticket at £68.30 or a standard Off-Peak at £95.40. They would be better choices.
You say we - how many in your party? If you are two, or one but under 30 or over 60 or have children with you, then a railcard is likely to save you more money. They cost £30 for a year but give 1/3rd off all prices. Given that you are presumably coming back as well you are likely to save even with just one train ride to Plymouth and back.

If you are booking a through ticket, Heathrow to Plymouth, which it sounds like you are, if you miss your connection at Paddington HEX to GWR then any ticket type (even an Advance) is automatically valid on the next train an hour later. At Heathrow get yourself in the front car. All main line platforms at Paddington are in a straight line across the concourse.
This is well within the Minimum Connecting Time at Paddington.

The bigger worry to me, is if the flight is on time.

National Express also run to Plymouth from Heathrow several times a day. I am on such a service tonight.
If their timings suit that is a more flexible option. If your flight is late and you have purchased the £5 flexible option you can catch the next bus.

through ticket, Heathrow to Plymouth, which it sounds like you are, if you miss your connection at Paddington HEX to GWR then any ticket type (even an Advance) is automatically valid on the next train an hour later.

Is that for any reason? If the Heathrow Express is on time but passenger misses GWR because of stopping for breakfast? Or plane was 2 hours late?

elaborating on Johnew52

If you are two, or one but under 30 or over 60 or have children with you, then a railcard is likely to save you more money. They cost £30 for a year but give 1/3rd off all prices.

You don't need to be under 30 or over 60 to get and use a Two Together Railcard, as long as you travel after 9:30 M-F (no time restriction on weekends or Bank Holidays (there are a couple of Bank Holidays in May when ticket prices will be high and trains crowded and sometimes engineering work on the tracks) ). Same £30 cost but valid for two named and pictured individuals who always are together when using the Railcard. https://www.twotogether-railcard.co.uk/

Re - Nigels post, timed at 2252- it was carefully worded for it being due to the Heathrow Express being late, not due to passenger default, hence even trying to be helpful in saying to travel in the front car.
The reasonable assumption was that the OP would be sensible in that regard.
At Heathrow Central the front car is the car the furthest away from the platform entrance.
I tried to cover for a late plane arrival by suggesting National Express instead.
Flix bus also now have one or two services a day on that route.
The OP will find fares via the Reading bus by using Station Code HWA, and a wider range of fares by using Station Code HAF (it doesn't matter that this code is for Terminal 4). There is a code for Terminal 2/3 But I'm working on mobile very late at night, and am struggling to trace it.
If you want a fully f!exible fare using Station Code HAF the cheapest option is to buy it as 2 tickets. Elizabeth Line on the day to Paddington, then a Super Off Peak Single at £68.30 Paddington to Plymouth, valid on any train after 10am except the 1603 and 1703.
I hope I have those 2 trains right- this is from memory.

Thank you everyone for your helpful advice . We’ve purchase a railcard Two together and will travel from Paddington to Plymouth on an off peak fare just in case there are any delays .

I suggest you also check what the price would be for a return (a 2 way ticket) - assuming that you are going back to London?