Are you using the same Apple ID on both the iPhones?
Have you enabled iCloud and turned on iCloud Photos?
When you say "synced my old phone to my new phone", what exactly did you do?
If I can understand you correctly; you want all past photos taken on old iPhone to new iPhone but do not want the new photos taken on old phone to get transferred to new iPhone. In this scenario.
On iPhone old:
Use Apple ID (Old)
Turn on iCloud Photos; Enable either Optimise Storage or Download & Keep Originals
On iPhone New:
Use Apple ID (Old)
Turn on iCloud Photos; Enable either Optimise Storage or Download & Keep Originals
On iPhone old:
Create New Apple ID & use the new Apple ID
End Result
On iPhone old: All past Photos and new photos when taken on this iPhone (Old) will exists
On iPhone New: All past Photos and new photos when taken on this iPhone (New) will exists