MultiBit actually released some tutorial videos for migrating to other wallets... This one shows how to export keys from MultiBit Classic...
https://www.youtube.com/watch?v=LaijbTcxsv8The most important thing to note, is that you DO NOT want to password protect the export file... otherwise, it will be encrypted and you won't be able to read the private keys!
Note: in the video they show how to import to Electrum, but when the video was made, they used v2.9.3 and in the latest version of Electrum v3.0.3, the procedure for importing keys has changed slightly... So, if you are going to migrate to Electrum:
If using v2.9.3, the procedure for importing, as per the video, is:
"File -> New/restore -> Standard Wallet -> Use public or private keys"
However, for new v3+ version, the procedure is:
"File -> New/Restore -> Import Bitcoin addresses or private keys"