Control+Shift+Power For current Mac keyboard layouts. Control+Shift+Eject works for older keyboard layouts that still include the Eject key.
This page explains some alternate solutions as well.
Also, if you have "Require password after sleep or screen saver begins" set to immediately in System Preferences -> Security & Privacy, it will ask you for your password after doing this.
First off, it's important to note that Apple's Remote Desktop is not Microsoft's Remote Desktop, and the two are not interchangeable. That said, you can still use various VNC products to connect from the Windows machine to the Mac.
From there, it's standard networking. If you want to leave the Mac Mini at home, you'll need to make sure that you can access the required ports, whether by port forwarding, or by an SSH tunnel/VPN allowing you to connect to your home network.
If you bring the Mac Mini with you, then you can easily create an ad hoc Wi-Fi network to connect them both too, in which case they will be on the same network, and you can connect that way. (You may need a monitor and keyboard attached to the Mac Mini at some point to configure this.)
Additionally, the Mac can automatically make any standard Ethernet cable into a crossover cable, such that an Ethernet cable connecting your Mac to your PC will create a network connection for the two machines, which, again, you can use as a standard network to connect from one to the other.
The network can be WiFi, direct point to point ethernet, point to point thunderbolt, point to point firewire or over a wired switched network of any of the above carriers.
Best Answer
There are many VNC applications that could accomplish this, and OSX has VNC support built in.
However, I really think you're better off buying a monitor, mouse, and keyboard for the Mac Mini. Get something used if necessary. You don't need to buy them from Apple - any standard monitor, mouse, or keyboard will work fine.