Flying
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 (£37 contactless payment) or pre-book it. 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.
Train
There are regular trains from London Paddington and London Marylebone stations, as well as from other UK destinations. It's a 20-25 minute walk to St. Anne's college and if you have luggage or it is dark we recommend this route.
Driving
We advise against driving if you can avoid it. Parking in central Oxford is difficult and it often simpler to leave your car at a Park & Ride (for example https://www.oxford.gov.uk/directory-record/95/pear-tree-park-and-ride). If you drive into central Oxford, beware also of traffic "filters", i.e. specific points that if you drive through will lead to an automatic fine.
Further information
A number of further options are listed the official Oxford page: https://www.ox.ac.uk/visitors/visiting-oxford/how-get-oxford.