You should only transfer the
.wallet file. This already includes the private key and therefor should NEVER be saved unencrypted in a server you do not control, or in a place which is not safe. You can import the
.wallet file on your other computer and do your bitcoin thingies from there. Make sure to not run both wallets at the same time.
In the main folder of Multibit this
.wallet file is stored. There is also an
Wallet-data folder. In there there are several backups of the wallet, and the private key. You can also export this private key, and import it on the other computer. In that way, the wallet will be generated (and will hold exactly the same amount of BTC after synchronising) and you can also do everything you want. Maybe that is what confused you.