London is connected through several airports and train connections. The best airport for Imperial College is Heathrow, but travel time from other London airports is also between 1 and 2 hours.
There is no need for rental cars. There are both black cabs and UBER.
There are several bike rental services (Lime, Forest...).
When arriving at Heathrow, you can either use the tube (Piccadilly or Elisabeth Line) or the Express Train to Paddington. The express train to Paddington is only marginally faster (about 15 mins) but more expensive. It leaves from the same platform as the Elisabeth Line so be careful to board the right train.
When arriving at the airport, there is therefore no need to buy a train ticket. On all London public transport you can use either an Oyster Card or any credit card. It is important that you stay consistent with the use of the card when tapping in and out, as it will automatically calculate the best tariff for you on a daily/weekly basis.
Transport for London (TFL) has an app. A very good app for planning your trips is Citymapper.
Some more information about public transport can be found here:
https://www.imperial.ac.uk/visit/summer-accommodation/tourist-information/transport-information/