Easiest way is by train. Heathrow is connected to central London by the Picadilly Line on London Underground (the tube) and by mainline rail services to London Paddington by the Heathrow Express service.
Plymouth is to the west of London on the coast in Devon and takes around 3hrs by Great Western trains from London Paddington.
So, if you want to save money then get the tube into London (costs about £6 but will take you an hour-ish). You will have to get Picadilly line to Earls Court and then change to District line to get to Paddington.
If time is your consideration then get the Heathrow Express which only takes 15 minutes and will get you right in on the platforms next to where you leave for Plymouth. A single ticket on that train is £13.50 if bought online in advance:
http://www.heathrowexpress.com/
(£14,50 if bought at the station)
Train fares vary wildly and not even the British understand them! Essentially the best way to get a train ticket is to book in advance online and collect them from the pre-booked ticket machines at the station. If you book more than a week in advance they are called APEX fares.
Go online to www.firstgreatwestern.co.uk or www.nationalrail.co.uk or www.trainline.co.uk to book tickets.
APEX fares are not available on all trains so if you are flexible about what train you get then you can shave a bit more off the price. I tried a return fare for April and it gave me £31.50 return ("round-trip" in US). However a single ("one-way") trip leaving later today would cost me £60.