You have some options. If you intend to record the on-board sounds from the Korg Triton, you’re going to need more than the USB MIDI Interface. Through a MIDI Interface you can only send data for a sequencer, like when a note is pressed and released, rather than the audio coming out of the keyboard. To capture the actual audio you’re going to need a digital audio recording interface, like the ones here:
Or, if sound quality isn’t a big issue, and the computer’s soundcard has a microphone input, you could just use that (but I wouldn’t recommend it–inputs on soundcards tend to be very noisy).
Once you have an interface, then you can record the audio output from the Triton using a digital audio recording program like Audacity. I would actually recommend you stick with Audacity. There are fancier recording programs on the market (Cubase, Logic, Sonar, etc), but I find Audacity gets the job done for simple projects, and you can’t beat the price.
Your other choice, if you don’t want to buy a recording interface, would be to use software synthesizers with the Korg Triton as your MIDI controller. This is where your USB MIDI controller comes in handy. Instead of hooking up the audio output from the Triton to the computer, you hook up the MIDI output from the Triton, so that a software synthesizer can generate audio output using the Triton as its input. I might actually recommend this method, as it will probably be easier for your students to structure their music this way.
If you want to take this approach, you’re going to need to get a decent DAW (digital audio workstation). I would recommend FL Studio, because it’s relatively inexpensive ($50-$150, depending on the version), quick to learn, easy to use, and has plenty of built-in sound generators and effects. Only drawback is that it is tailored to make electronica and hip-hop, so it is laid out in a way that might not helpful for someone trying to make, say, classical music, but I would say it is flexible enough to accommodate the different styles of music your students may want to make for their films.