After the launch of MultiBit HD, I decided to move from MultiBit to Coinbase. What is the cheapest way to transfer all my money from one wallet to another? Can I somehow export/import my key/wallet (if yes, how) or the only way would be to send money and pay transaction fees (if yes, should I transfer them all at once or in smaller chunks)?
You should reconsider your wallet switch. Mind that Coinbase can fully control your funds and it could lead to a similar fiasco we have seen in the past (Mt.Gox, etc.). Check out the web wallet section of bitcoin.org.
Export and import (if even possible) would expose your private keys or mnemonic seed and should therefore be avoided.
A better option would be to send your coins from Multibit HD to Coinbase. It would be one transaction and the required fee would be not very hight (~0.0003 BTC).
You can export your wallet to Blockchain.info, and then send the bitcoin to Coinbase for 0 fees. This, by definition, is cheapest. Please notice however that if you have many small transactions ("dust"), your request might take a long time to confirm, if at all.