Trains from London. Two major stations connect London and Cambridge station: London King's Cross Station and London Liverpool Street Station. The trains from King's Cross are faster, and the trains from Liverpool Street are cheaper. You will need to buy tickets for these services, which can be done at the station or online. (trainline.com is a good website for buying tickets).
Flying to London Airports.
Heathrow. There are regular Elizabeth Line and Piccadilly Line trains from Heathrow into London. These are part of
Transport for London and you can pay by contactless card or Apple Pay/Google Pay, etc. If you land at terminal 5, the services from there are less frequent but get on the first train to terminal 2/3, and you might get something quicker (trains between the terminals are free). The Heathrow Express service also takes you to London Paddington, but this is significantly more expensive.
Gatwick and Stansted. Regular train services go directly from Gatwick and Stanstead Airport to Cambridge; again, you will need to buy a ticket for this either at the station or online.
Luton. Luton can be a difficult airport to get to or from. A National Express coach goes to Cambridge, or you can get a 100 bus to Hitchin and get a train the rest of the way to Cambridge. For National Express coaches, you can either pay the driver on the day or get a ticket in advance.
London City Airport. The easiest route is to take the DLR westbound to Bank, then catch the Northern Line north to King's Cross and catch a train to Cambridge. The DLR is part of Transport for London, and again, you can pay by contactless card, etc.
Getting to Murray--Edwards College. Once you're in Cambridge, you need to get to Murray--Edwards College. In most cases you can walk, but the city is quite small so taxis are fairly cheap. Uber is available, but Panther Cabs is also a good choice. There is a taxi rank directly outside the Cambridge railway station. Once you get to Murray--Edwards, head to the porters' lodge (reception), and the porters will be able to direct you.
You have many options in Cambridge; here are just a few:
University Arms Hotel, in central Cambridge https://universityarms.com/
Hotel du Vin, in central Cambridge https://www.hotelduvin.com/
Ibis Hotel, by the station, https://all.accor.com/hotel/8548/index.en.shtml
The Hilton, central Cambridge https://www.hilton.com/en/hotels/stnhchi-hilton-cambridge-city-centre/
The Graduate, near to Peterhouse and the river https://graduatehotels.com/cambridge/
There is also the standard assortment of Premier Inns/Travel Lodges/Holiday Inns, which are a bit further out but consequently cheaper.