The procedure for moving private keys from one wallet to another, if that is what you want to do, will vary based on the specific wallet, but if you had something similar to a bitcoin core wallet, the general commands would be first to use either:
dumpprivkey <address>
or
dumpwallet <filename>
The first would give you the private key for a single address that holds bitcoin. The second would put all of your private/public key pairs into a file. Both of these would first require you to unlock your wallet if you have locked it with a passphrase.
Once you have the private key(s), you would do the reverse in the new wallet:
importprivkey <key>
or
importwallet <filename>
That would bring over either that one address or all of the addresses into the new wallet.
If you have been doing a lot of transactions in your old wallet, you may have bitcoin on many different addresses. You will need to know which addresses have bitcoin balances to know which ones you need to transfer.
Of course, there is also the much easier method of sending the bitcoin from your old wallet to an address in the new wallet, as long as you don't mind the transaction fee and waiting for the transaction to be confirmed.
