For the past ever, I've been on my parents' Apple ID, making purchases on their credit cards and sharing any iTunes gift cards I redeemed with the family. However, I recently created my own Apple ID, complete with my own payment method. I'm tempted to upload every single gift card I have to my account. Is this doable, and are there any limitations to doing it?
Yes - stacking gift cards is not an issue. You will just carry a larger balance and less risk of losing the cash.
You lose the flexibility to redeem them on another account.