If your photos are in iCloud, you cannot prevent them from being transferred to your new iPhone when you sign in with your Apple ID, and thus also your iCloud. The only solution is to deactivate iCloud Photos, however this will also mean that your photos that you take on your new iPhone will not be saved and transferred to iCloud.
You can disable iCloud Photos on your new iPhone here:
Settings → Apple ID → iCloud → disable Photos
Everything you have in iCloud will be synchronized to your new iPhone when you sign in with your Apple ID. If you don't want it, disable Notes, Safari, Calendar, etc. in the same menu; Settings → Apple ID → iCloud.
Your apps will not be automatically downloaded to your new iPhone unless you choose to do so. In the setup, just select: Don't Transfer Apps & Data.

You can find more useful information here:
Set up your iPhone or iPad - Apple Support