Can I transfer my Minecraft world from one iPhone to another? How?

I have Minecraft Pe and i was using it on my old device, I'd like to continue the world that I've built on my new device, how do I transfer it?

The only way to do this would be to back up the old iPhone to a computer running iTunes & then restore that back-up to your new iPhone. That's about the best (& only) way to do so.

Please be aware that Minecraft has evolved beyond the "Portable Edition". With the release of the Bedrock Engine (after the Better Together update), it's now the considered the "standard edition" of Minecraft (where the old version is know the "Java Edition").