Can you choose certain things to transfer from old iPhone to new?

So when you back-up everything to iTunes and transfer the data to a new phone, is it possible to only transfer say music and contacts and leave photos for example?

Maybe

Contacts should automatically transfer from your sim, and music from is from itunes on your pc so just sync the new phone from there, so ultimately don't transfer anything at all

I think it is easy to solve your problem.
To transfer files from old iPhone to a new one. I do think you can try iCoolsoft iPhone transfer, it can transfer files among any iOS devices, like iPad to iPhone, iPhone to iPad, iPad to iPod. Also, it can transfer Photos, Music, Videos, and more between iPhone and computer; Backup iPhone Contacts and messages to PC local disk; Sync iPhone files to iTunes for backup just with few clicks.