Obviously there is 'goodbye' which is more formal. You can also say 'see you soon.' There are other expressions like 'take care' which are also used to say goodbye. We also use foreign language goodbyes in British English such as 'Ciao', 'Adios', and 'Au revoir'.

Oh yes, there are many other expressions that people use when they are parting. For example, if they have a plan to see you soon, tomorrow, next week etc., they might say: : "Talk to you later," or "Until tomorrow." If there's no plan to see you soon, and they want to get together again, they might say: "Don't be a stranger," or "Let's stay in touch." There are a lot more ways that people say "goodbye." You could probably find more if you spend some time listening to TV shows; just focus on how the different characters say goodbye in different contexts. I bet that would be a fun research project!