The closest airport is London Heathrow. The simplest way of getting to Oxford is The Airline coach service. You can purchase a return ticket from the driver. It stops at Terminals 2, 3 (Central Bus Station) and Terminal 5, while from Terminal 4, you should take a free train to Terminals 2 & 3.
During the day, the service runs every 20 minutes and it takes about 1h40 minutes to central Oxford. Take the bus to its last stop ("Gloucester Green"). St. Anne's college is a 15-minute walk.
The Airline also connects Oxford with Gatwick airport, running once an hour.
Some people like to use Birmingham airport, which has a direct train connection to Oxford.
A number of further options are listed the official Oxford page: https://www.ox.ac.uk/visitors/visiting-oxford/how-get-oxford.