We are flying into Heathrow to Terminal 1 and will need to go to Paddington Station to take a train to Oxford. How do we get there exactly.

When you leave T1, follow the sign for Heathrow Express/Train to London. You'll probably need to take a lift down.
MAKE SURE you buy tickets before you board. Also MAKE SURE you're in the right platform (one for London, NOT one for Terminal 4).
You can take Heathrow Express (direct, more convenient but more expensive option), or Heathrow Connect.

I'd recommend taking The Airline bus direct from Heathrow to Oxford. It takes 70-90 minutes. Much easier than going into London to take a train, IMO.

To JoeTro, I love the bus idea do you know where we go when we come out of the terminal to catch it and how much it will cost approximately. Thanks

Fares are 20 pounds return.

http://www.oxfordbus.co.uk/heathrow.html

For those who aren't familiar "20 pounds return" means "20 pounds round trip".

As someone who normally suggests not using the Heathrow Express to get into London as it then means a transfer and a taxi to get to a hotel -- I must say it is the BEST way to just get to Paddington in this case.

But I'd think the idea of a bus direct to Oxford that only takes up to 90 minutes sounds a lot easier than going to Paddington and they getting another train to Oxford.