Most of the sounds, such as the humming noise heard from outside the TARDIS (the first indication in the opening episode "An Unearthly Child" that the TARDIS was not just a police box) & the buzzing noise of the doors opening or closing, were made electronically. Many of the beeps heard from the control panel were also made electronically.

The sound of the Cloister Bell was made by striking a gong that was partly submerged in water.

The most famous of the sounds, the grinding noise heard on takeoff & landing was made by recording a door key being scraped along a piano string (made of coiled wire) & then modifying that recording -- altering its speed, reversing the tape, & so on. There have been many versions of this sound, with different processes used to modify the basic recording. There have even been other takeoff & landing sounds used. The article The Sound of the TARDIS: A History goes into much more detail on this subject.