How do I obtain the idols?

Much as lunboks described, all the idols aside from the first must all be bought from the Lost and Found shop. There is a total of 10 idols. There are 7 to be bought in the first playthrough and you need to upgrade the idol shop to get some of them and there are 2 that can be purchased from the New Game+ Lost and Found.

The effects of all the idols can be found on the wiki:

Idols to each of Caelondia's deities may be activated at the Shrine. Activating an idol increases the rewards the Kid receives for defeating enemies; however, each idol also makes enemies more difficult to defeat. This allows for a high level of user customization of the game's difficulty. Idols that are turned on will remain on for both the story and Who Knows Where arenas. The online leaderboards for Who Knows Where do not track scores, but rather the Idols that a player had on when they completed all 20 waves.

Idol Name                               Description
Hense (Goddess of Pain and Pleasure)    Foes are more ferocious
Acobi (Goddess of Oath and Abandon)     Foes drop a live grenade when defeated
Lemaign (God of Hope and Despair)       Foes' attacks slow down the player
Pyth (God of Commotion and Order)       Foes are quicker, both with movement and attacks
Jevel (God of Health and Atrophy)       Foes are more resilient
Yudrig (God of Impulse and Bravery)     Foes cause damage on contact
Roathus (God of Thirst and Plenty)      Foes never drop Health Tonics or Black Tonics
Micia (Goddess of Loss and Longing)     Foes have regenerating health
Olak (God of Chance and Whim)           Foes occasionally turn to air, making them invulnerable to attacks for a short period of time
Garmuth (God of Purpose and Folly)      Foes are more capable of deflecting attacks

According to the wikia, the rewards Kid receives are "a passive XP and Fragment boost".

I'm pretty sure Pyth Orchard is the only level that has an idol in it. I don't think Pyth even counts as a collectible, since you have to visit the Pyth Shrine to end the level.

All the other idols are obtained by buying them, after unlocking them first by doing things like upgrading the Shrine or starting a New Game+.