Aside of Zenkai, which is one of the reasons why Goku increased his power very fast, another reason is they started to use training under heavy gravity.
Before the battle with Goku, Goku trained with King Kai in a planet with a gravity 10 times stronger than Earth's. That made him able to increase his power very fast. Also after that, Goku made Bulma's father to built a device which created an artificial gravity field of 100 times stronger than the one of the Earth. With that device and the Zenkai from the battle with Vegeta, Goku was able to raise his base power level from 9,000 to 90,000. Vegeta also demanded Bulma's father to create a device with at least created a gravity field of 300 times stronger of the one on Earth.
Moreover, Goku in particular learnt from King Kai a technique called Kaioken. With Kaioken he was able to multiply his power level. When he applied kaioken times 2, he was able to raise his power level from 90,000 to 180,000, in the battle with Ginyu. But he was able to raise kaioken even up to times 20.
In his battle with Freezer, he was already 3,000,000 in base form due to a zenkai he got after being beated almost to death and having to recover in the medical machine. So applying a kaioken times 20 over that, it made him to have a power level of 60,000,000 during a short period of time.
And aside of that, saiyans have transformations which apply to them multipliers. The super saiyan transformation has a multiplier of times 50. So when Goku transformed, a times 50 multiplier was applied to his base form power of 3,000,000 during that time . Meaning he got a power level of 150,000,000 when fighting with Freezer in super saiyan form.
The power levels of Goku I mention, you can see them stated in this list.
List of power levels