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If I buy a second computer, can I just access it remotely through my laptop, or do I need to get the monitor and mouse as well?
The computer will not be a server, just a regular desk top.
One of the issues you face for setting it up is its IP configuration. Are you connecting to your network and the internet through a router? Do you want to be able to connect to either machine from other remote locations, or just one, which one? DHCP or Static IP?
Another issue is firewalling the remote PC.
One little misstep with either the IP configuration or the firewall via remote access and you are toast without an onsite presense, monitor and keyboard.
It is unlikely your laptop will serve as monitor, keyboard and mouse for the desktop machine.
It is possible the desktop machine, once properly configured will never need you to connect a monitor to it.