The Easter Eggs can only be found specifically in one exact race in Adventure Mode. I have made a video listing all Easter Eggs and I will also write and describe it the best I can for those who prefer text.
(Though video is much more easier, for clarity and visually showing over trying to explain)

I've listed the eggs in order of their appearance in Adventure Mode as followed:

Easter Egg 1 - Marsh 6 - Mushroom Grotto - 3 Lap Race
This can be found when the cave opens up and you start too see mushrooms. It's to the far left hidden behind a small rock.

Easter Egg 2 - Swamp 4 - Tiki Temple - Drift Attack
To see this Easter Egg, you must take the second shortcut which leads to the portcullis which opens up. (break through the barricades and go up the raised path).

Easter Egg 3 - Jungle 10 - Buccaneer Bay - 3 Lap Race
At the start to your right, behind the cart you'll notice an Egg hidden behind the grass hill and wall. Next to the windmill.

Easter Egg 4 - Space 1 - Red Planet - 3 Lap Race
In the first section where you have no gravity, it'll be to the far right, where the first shortcut is in the jumping section. Slow down as you can go too fast and get reset on track before you can smash into it.

Easter Egg 5 - Volcano 6 - Fire and Ice - Time Trial
This egg is after the first cave, and you have to circle back on yourself, it'll be on the ledge next to the lava. You can also see this egg during the opening cutscene of the race.

Thanks a lot, I put a lot of hard work and effort into this.

https://youtu.be/bPFWm6Xyv7E