If you want to use utilities that are already present on Mac OS X systems, you can use Secure Shell (SSH), which is a more secure means to log into a remote system than telnet, since SSH connections are encrypted. First you would need to configure her system to enable it to function as an SSH server. You can do that through System Preferences by going to Sharing and checking the box next to Remote Login to enable the SSH server service. Then on your system, you would need to open a Terminal window (you will find the Teminal program in Applications/Utitlities) and run the ssh client program on your system. You would use something like ssh [email protected] if the userid on her system was jdoe and its IP address was 192.168.0.3. If you are accessing her system remotely, she can give you its current IP address by visiting a site such as www.whatismyip.com, which will display the IP address from which a visitor has connected, so would tell her the IP address to which you would need to connect. For the userid when logging in by SSH, use the "short name" for the account, which can be found by going to System Preferences, Accounts, then clicking on the lock icon, and then selecting the account.
An SSH connection will give you a command line interface, which is what you get with telnet as well. You won't get a graphical user interface (GUI) which may be more familiar to you, but you can move files and directories from a command line interface and edit text files, though you will need to use a text editor, such as vi or nano, which should already be on her system. But you can configure her Mac to function as a VNC server to get a graphical interface for remotely managing her computer. You can enable that functionality through System Preferences by selecting Sharing then Remote Management. You could then use the VNC client software on your system for remote screen sharing. You can use VNC from your system by opening the Safari web browser that is present on OS X systems, but then, instead of typing http:// followed by an address of a website, you specify VNC as the protocol followed by the address of the remote system. E.g. vnc://girlfriend.example.com or vnc://192.168.0.3.
You can also enable both SSH and VNC connectivity and "tunnel" the VNC connectivity over SSH for better security. You would need to tunnel port 5900. Note: for you to be able to access her system remotely, i.e., from outside the router/firewall at her location, you or she would need to configure the router/firewall to allow remote connections on ports 22 (SSH) and/or 5900 (VNC) through to her system. If you do so, you must ensure that the account or accounts on her system to which remote access can be gained have strong passwords. That should be true before you enable any type of remote access method to her system, since there are nefarious individuals throughout the world constantly scanning the Internet for systems with weak passwords which they can compromise either to look for credit card or other financially useful information or to use to hide the true origination point for their attacks on other systems.
TeamViewer, which has already been suggested, will be simpler to get working, e.g., you won't need to worry about configuring the router/firewall on her end to allow remote connections through to her system, and is free for personal use, but the above are options that will allow you to use software that should already be present on both systems, if you both have Macs with OS X, and can commit more time for getting remote access functioning.