Good news that you don't have a car - it's the last thing you need in London.
Either Oxford or Cambridge would make a great day trip. I wouldn't do both but there's little to choose between them: Oxford has the Tolkien and Inspector Morse links; Cambridge has the glorious Backs (Google them) and Kings College Chapel.
For Oxford, get London public transport (tube or bus) to Paddington Station and catch the train - takes about an hour.
For Cambridge, get yourself to Kings Cross station in London and catch the train - takes about an hour. There's also a slower service fromLiverpool Street station.
As these are relatively short journeys I don't think there's any cost advantage to pre-booking. I believe each would cost in the region of £20/£25 for a return (round-trip) ticket per adult.
Details of train times and fares can be found on www.nationalrail.co.uk - use London (All) as your starting point.
I don't know anything about golf but this article (from four years ago) looks like a good starting point, and gives public transport details too. "Inside the M25" means inside the giant outer London motorway.