When starting a new cup, you will be(usually) near last place. And usually, you get the chance to get triple mushrooms(The item I prefer to get.) And I was wondering if it's better to use it almost instantly or save them for shortcuts.

Save them for shortcuts, there's basically no reason to use them on straights, except in specific circumstances. You can also dodge blue shells with them.

I guess it really depends on what situation you're in. if you just got it off the bat, save them, in case you get hit or for short cuts.

if 2nd place has reds, and you can't utilise the track you're in to destroy them: use the mushrooms, getting hit is 3-4 seconds easily lost
if they don't have reds, utilise them on the best track shortcuts, however if game allows, save 1 mushroom for as long as possible to protect against blue shell, although if none appear by mid lap 3 then you should use it

I mainly use 2 for shortcuts and save 1 for a possible blue shell. If no blue shell comes during the entirety of the race, I use it at the very end of the race for a slight speed boost. It's also important to remember that you can receive mushrooms in 1st place, so I usually spam my items until I get one so I can dodge.