LogMeIn is totally safe. It uses SSL protocol (Secure Sockets Layer) which encrypt data when sent throughout internet . Furthermore, you need two passwords: LogMeIn password and the computer's password.
LogMeIn offers you an easy-to-use and intuitive interface which lets you setup some options related to remote control, security or net configuration.
The free version offers a lot of features, but, for example, file transfer or remte printing is only included in the version PRO.