To become a founder of an alliance or a member, you have to upgrade your firestation to level 4. Afterwards, you can build the radio station next to your firestation. With a radio station you can start your alliance career.
Open up the alliance menu by clicking on the alliance button on the left. To become an alliance member you have to apply for membership first. Send your application via the profile of your chosen alliance. You can find alliances by searching for them via the alliance search, or simply by clicking on scores and find them this way. If you wish, you can also apply for membership at several alliances at once. After that, you need to wait and see, if, or when your membership application will be accepted.
Using the pencil in the alliance menu, you can establish your own alliance.