I just use headphones straight out of the DI (Behringer U-Phoria UMC204HD) if recording. I even use that to play the GT lessons through the same headphones, so I get the sound from those and the guitar/pedals. Studio monitors would work the same way, but I prefer a good set of headphones that can handle the frequency ranges (never go cheap on audio gear unless you're sure it'll do what you need, or have no choice due to budget limits).

Other times it's headphones straight into the Iridium if I want to be quiet.

Purists will still say it's no substitute for the real (and expensive, high maintenance) amps, but I love the sounds that the Iridium can knock out - and it fits on a pedalboard. The specs state it needs 500mA with headphones, though I measured its power consumption which was 300mA (I still play safe and let it have a full 500mA to itself).