What can I do with Bitcoin Core to see my current balance?
Bitcoin Core incrementally shows transactions you received as it processes blocks, so if you received the transactions a while ago they should pop up soon. You will not be able to independently verify the validity of your transactions without fully synchronizing.

If you just want to check the balance of your addresses it might be sufficient to look up the addresses on a public block explorer. The JSON-API gives you various methods for retrieving your addresses from Bitcoin Core. Once you have the addresses simply check their balance using a block explorer.

If however you'd like to move to a different wallet, the backupwallet API should give you a copy of your wallet that other wallet software can import.