Right now, it's not possible to merge two trips in Tern. The best way to do this would be to save one of the trips as a trip template by clicking the action button on the itinerary you'd like to merge with another trip and clicking "Save itinerary as template".
Then, navigate to the trip you want to merge it with. Drag in the "Itinerary Library item and select the itinerary you saved in the previous step.
That will add the itinerary to the board and effectively merge the two trips. Note that other information about the trip (e.g. traveler information, etc) won't come over as a part of this process. Authorizations also don't carry over. Saving an itinerary as a template copies the itinerary content only — signed client authorizations and payment authorizations stay with the original trip. You can't move an authorization by downloading and re-uploading the document; an uploaded file is just an attachment and won't let you charge the client. Once you've dragged the items into the trip you're keeping, have the client authorize again on that trip so the payment authorizations are tied to it.