Options getting from Heathrow to Oxford

The Oxford Bus Company runs a bus from Heathrow Central Bus Station and Heathrow Terminal 5 directly to Oxford. It is called the Airline. Info at http://airline.oxfordbus.co.uk/.

cheapest I know of does have a bit of maneuvering. First catch the tube to south  Kensington (Direct line from the  Heathrow Airport) then change to the district line (green) and go one stop to  Victoria. From there you can grab a bus. Pending if it's an express or not you can get a ticket  anywhere from £5-£15. this might take you over 2 hours .  other option is train. Express to  London Paddington Railway Station (PAD) then a direct train to Oxford. you would shave at least half hour but will be more expensive.

National Express Coaches will take you from Heathrow Airport to Oxford I believe: www.nationalexpress.com.  I think it's about £25.

There is a direct bus from Heathrow Airport to Oxford City Centre. It's as quick as the train because you don't go back on yourself, and much cheaper.
You can see the route at: http://www.rome2rio.com/s/London-Heathrow/Oxford

Here are some options:
1)
Take the Heathrow Express to London Paddington Railway Station (PAD) . One way journey is about 24 pounds (coach). The trip is about 15-20 minutes.
Take the train from London Paddington Station to Oxford. One way journey varies from 15-23 pounds (coach). The trip is about 50 minutes.
2)
Take the Heathrow Express to London Paddington Station. One way journey is about 24 pounds (coach). The trip is about 15-20 minutes.
Take the underground from London Padding Station to London Victoria Railway Station (VIC). The trip is about 10 minutes and will cost 4-5 pounds. You'll have to come above ground since the bus stop is opposite side on the street.
Take the Oxfordtube bus (24 hours a day, every 10 minutes) to Oxford. The fare is 14 pounds one way and 17 pounds roundtrip, so RT is definitely the better value here. However, the journey takes a little over two hours. For just 1 pounds less than the train one way, I'd recommend the train.

If you go down to the "busses" then there is a direct Heathrow Airport to Oxford bus, just ask any desk agent when you get there.  Same goes for Gatwick.  Great busses, free WiFi.  They're around £23 for a one-way ticket. If you want more info: http://www.oxfordbus.co.uk/

Take bus no. 140 from stop 19 at  Heathrow Central Bus Station towards Harrow Weald. Stop at Hayes & Harlington station. It will cost you around £2.40 cash or £1.45 with an oyster card. The journey takes aprox 20 mins. Then from there you can take the train straight to Oxford for approx £17.40 single. The journey takes 1h30mins and the service is operated by First Great Western. This is the service that leaves from London Paddington. Doing it this way saves you the trouble and money going back to Paddington.

Heathrow Express will get you to  London Paddington Railway Station (PAD) (~15 minutes - see https://www.heathrowexpress.com) and you can get direct train to Oxford (~1 hour - see http://www.firstgreatwestern.co.uk) from there.

I think coach (bus) might be your best bet! There's a line that runs just from Heathrow Airport to Oxford and back.  Hope you have a great time! 
http://airline.oxfordbus.co.uk/timetables/

To get to Cambridge, I always take a coach with National Express. It takes perhaps 1/1.5 hours longer than car/train, but is super convenient. It goes point-to-point, and they're usually ok with my 2 suitcases without needing to pay an extra baggage fee. You can then get a cab from the coach stop in Cambridge to your final address.
I imagine the same applies to getting to Oxford.

There's a train station nearby in Hayes And Harlington (I don't know how much a taxi taxi costs from Heathrow Airport, but there are also regular buses), you can get a train from there to Oxford with no changes in about an hour and a half. Take a look at http://nationalrail.co.uk for train times and prices etc :)