Copy/paste from a previous post of mine.
It's all luck based, so there isn't really any trick. If you have consumables to burn that can give you a rainbow button, it seems to increase your chances, it's just very 'expensive' to do.
For those that don't know, as you drag items into the slots and/or change the amount to give her, the button at the bottom will change from Bronze, to Silver, to Gold, and finally Rainbow or Diamond or whatever you want to call it. The colors define what type of acorn, and I think if you're closer to the next tier, it's a chubby acorn, vs a skinny one.
Every item has a different 'weight' to it towards getting you to the next tier, so experiment a little to see what you can use.
The items you put in all have a different 'weight' to them. You'll notice that putting more precious things like card reveals or EX wildcards will be worth more than pink wildcards.
From what I can tell, EX cards are very heavy, Orange tickets are very heavy (duh, it's real money) and the Girl reveal and Glasses reveal are pretty heavy. From there its probably Silver pot, silver boob, r gacha, then pink pots and boobs, and 5x golds. 3x and 1x golds and silvers I think are about the same, and then the rest is just throwaway junk.
I'm not too sure about the radars or the silver/gold keys. The pink/orange boss potions and the pink gacha tickets/pink cards are all pretty low because they can easily be bought with a relatively low amount of in game gold.
Your chances of getting better stuff seem to increase when your button's 'color value' goes up, but since it's luck, you can still get junk from a gold acorn vs a brown one. Same thing with the nest-specific cards. People have found the gold card in a bronze acorn, and pinks in a gold 'rainbow' one. Just put stuff in, wait, and repeat.
The higher ranking acorns do take longer before you can open them. I think the lowest is 9 min, and a rainbow is about 5 hours. If you have any champagne bottles from a past event or if you buy them with money, they will immediately stop the timer and let you get the prize.