No need to be confused
Checking your journey on
National Rail gives the route via Woking as the fastest (also will be cheapest). This involves a
bus to Woking, then train to Gillingham, so you will have to buy tickets to Woking on the bus (or at the airport bus station ticket office). There doesn't seem to be any railcard discount on the bus anyway, so you won't lose anything. When you get to Woking you can buy the railcard and tickets there.
As
MikeWh pointed out, from Heathrow there are only trains to London. If you did want to get the train to Gillingham from London, you will need to use Waterloo station, and I'd recommend using the Underground to get there - it will take about the same time as getting a train into Paddington station and crossing London, but would be much cheaper. You can get your railcard and tickets at the same time from the ticket office at Waterloo station.
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our full travelplan is: Heathrow T5-Gillingham-Harpenden-Oxford-Heathrow T5.
I would recommend (assuming you're travelling "off-peak" - your railcard is valid after 10.00 in the week):
Heathrow T5 - Woking - Coach link, £10.50 single. Buy the Network Railcard at Woking.
Woking - Gillingham - Off-Peak Return, £24.90 (per adult, with railcard discount)
Gillingham - Harpenden - buy a
Woking-Harpenden Off-Peak Single (£15.80). Use the return half of your Woking-Gillingham ticket to Woking, then the Single (it includes the Underground for crossing London). The train you are on needs to call at Woking (they all do) but you don't need to get off.
Harpenden - Oxford - Off-Peak Return, £29.95. Again, this includes the Tube for crossing London. Make sure you get a fast train from London Paddington to Oxford (usually at XX22 and XX52), the stopping trains take ages.
Oxford - Heathrow - buy a
Hayes & Harlington - Heathrow Anytime Single, £5.20 (or £3.45 if a weekend). Use the return half of your Oxford-Harpenden ticket to travel to Reading, then change to a local train to Ealing Broadway. Change again at Hayes & Harlington for the Heathrow Connect service to Heathrow.
Alternatively for the last part you could use the return ticket to Reading and then take the RailAir coach to Heathrow - this would be quicker and more convenient, but more expensive (the coach is £17 single).
To check train times, use
National Rail.
If you want to buy tickets online, use one of the train companies as they won't charge any extra fees. For "normal" tickets though it isn't any cheaper to buy online. Also note that Single tickets and the outward part of Return tickets are only valid on one day, so if you're buying in advance make sure to specify the date! The return part is valid for a month with these tickets.