How would I get my ssh keys to the remote server from a new machine?
To me, the easiest way seems to be either sharing private keys from a different machine, or having some way to deterministically generate keys from a password or keyphrase, and the latter seems more secure to me because I don't have to trust a middle man to do the transferring.
Generate a new private/public key pair on the new machine and then copy the public key (from the new machine) into the authorised keys (on the server) using a login that already has access (e.g. from the old machine).