Make sure you're signed in with the same Apple ID on both devices. Wait for a message about merging contacts to appear, and then tap Merge. The easiest methods for transferring contacts use the features that come with the iPhone, such as iCloud. Because iCloud can keep data synchronized across all your devices, using it to sync contacts from one iPhone to another makes it an easy option.
Beyond the contact synchronization process from the last section, iCloud can also backup your iPhone data and then restore it to a different iPhone. If you just need to make a couple of quick transfers, here's how to transfer contacts from iPhone to iPhone using Airdrop.