I just bought a new iPhone 6s, and want to move all the contacts from the old one to the new one. Can iTunes or iCloud help me do this, or are there any other ways?
Yes either iCloud or iTunes can do this.
The overview is that you take a backup from the old iPhone and restore during setup on the new iPhone.
Apple has a support page for this: Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch
When my daughter filled her iPhone4 (yes they still exist!) with photos so that she was unable to update the operating system, I connected the phone to my computer and ignored the request to fire up iTunes and simply looked at the phone contacts as if it were just any old external drive. I copied a chunk of photos folders without changing their coded names, deleted them on the phone, disconnected iPhone, updated the OS, then reconnected computer to put back most of the folders in the same place without further interference. She needed a clear out anyhow. Works fine.