This helpful video by TWD98 shows almost all shortcuts that doesn't require mushrooms in Mario Kart 7.
However, some requires a lot of practice and courage, while some require certain kart parts, for example, Monster Tires will prevent losing too much speed while driving on inaccessible areas such as grass, sand for a short distance.
For me, I never knew that we could fly under the bridge in Neo Bowser City, and I never noticed that ramp in the lava in Bowser's Castle (New one, not the GBA one) before.
Hmm...I've known some of these, and I use a modified version of the Neo Bowser one already (albeit not as efficient) but some of the others were interesting.
I knew all of the shortcuts except for that Neo Bowser City one, the one where you could jump onto the wall on Bowser Castle 1 without using a mushroom, the first one without using a mushroom, and the Mushroom Gorge one without using a mushroom.