It is a two stage process, buying the ticket (which you have already done) and getting a boarding pass by checking in.
You can either check in online in advance (it will say when check in opens), and print your pass with a code, or use their app and have an electronic version on your phone, or do it on arrival at the airport (either through a self service machine, or at normal check in desks).
To check in, you need the code (few letters and numbers) that you were emailed with your purchase receipt. For hand luggage only I wouldn’t bother to queue at check in desks
At Heathrow T5, departures are on top floor so if arriving by train take lift (quicker than escalators), then if haven’t already checked in use a self service machine for boarding pass, then go straight to security, the domestic gates are on the left (I think you have to go down a level in departures from memory)
If you have business class, there is a lounge on the right after security with free refreshments and more comfy seating than in the main departures lounge.
Seat reservations are free within 24 hours of the flight (longer if you hold various frequent flyer tiers), and free on First class (but that isn’t available on domestic flights). So don’t waste your money paying for these if you are travelling on a short flight on your own. You can choose your seat when you check in (it will offer one, but can move it to any that are still available), but this needs to be done before completing the check in as Seat number will be on boarding pass
For domestic flights you don’t need a passport, but they might ask for photo ID, so often easier just to carry your passport than find something else.
