I'm not exactly sure how I'm supposed to combine everything.
Oh, at any rate... the setup you mention can be connected to any computer, such as a laptop with a simple audio jack?
Audio interfaces / breakout boxes are typically connected by USB, so they will work with either a desktop or a laptop computer - you plug the mic and the headphones to the breakout box and connect it to the computer, then run mixing/recording software that comes with the box.
There should be at least one XLR input with +48V phantom power, and a 1/4" TRS jack for headphone output (or a 3.5 mm mini-jack).
If you want a complete starter kit, there is
Focusrite Solo Studio Pack (Gen 3) - but the included headphones and the electret-condenser microphone are not particularly good (though probably better than cheap USB-connector mics with integrated preamp and ADC).
There is also
Rode AI-1 Complete Studio Kit, which includes a large-diaphragm studio microphone, an audio interface, and a shock mount - so you only need to add a mic stand and monitoring headphones.
For top sound quality, look for pro-audio interfaces from Focusrite, Presonus, MOTU, Steinberg etc., large-diaphragm condenser mics from
Newman, AKG, Rode, Audio Technica etc., and closed-back studio headphones like AKG K271, Beyerdynamics DT770 Pro, and Sony MDR-7506 - but that combination could easily cost you $500 and more...
Are "closed-ear monitoring microphones" actually "headphones"?
Yes, that's "headphones" of course. They just have to be 'closed-back', noise-isolating type - this is critical for studio monitoring work, as opposed to music listening.