Where does the litecoin wallet store coins?
On Linux, the default data directory is ~/.litecoin/. This is where the wallet, blockchain, configuration, etc. files are stored.

You can backup your wallet from File > Backup Wallet. This is the part that contains your private keys. See the pic:



You should encrypt with a secure passphrase (in Settings > Encrypt Wallet) before exporting your wallet. If you don't, the private keys in your backup will be unencrypted. As long as your wallet has been encrypted, your backup will be encrypted and you can safely* put it on Dropbox, Google Drive, email it to family/friends, etc. for safe-keeping.

*As safe as your passphrase is, so make it very high quality.