Can I make a new Apple ID without losing everything on my phone?

You won't have to reset your phone. Sign out of the common Apple ID, to include FaceTime and Messages, create a new ID for yourself, and then sign back in. The contents of the phone should remain intact. Purchases are tied to the Apple ID that was used to purchase them, so they would have to be obtained by the new Apple ID. Depending on your ages, you might consider Family sharing, which would allow shared purchases. Your parents could set this up as the organizer.

Apple ID – Create

Family Sharing

Family Sharing -What types of content can I share?

To answer your first question, see this article:

Change your Apple ID - Apple Support

(Basically, you just want to follow step 1 in the "Change your Apple ID" section. You will be creating a new Apple ID, not changing it.) Then see:

Create and start using an Apple ID - Apple Support

and especially the last "If you want to sign in with a different Apple ID" section (and the link therein).

For your second parenthetical question, be aware that Apps & Songs purchased through the App & iTunes Store are permanently tied to the original Apple ID and cannot be shared or transferred to another Apple ID. However, you could use Family Sharing with your sister (and still have separate Apple IDs) to share those purchases:

Family Sharing - Apple Support

Set up Family Sharing - Apple Support

Share purchased content with Family Sharing - Apple Support