Okay, I am going to give you a simple practical answer. Let's start with the sound. If you want to record live performance well, then you basically need good signals into your recorder (obviously, right?). And that recorder is often these days a program on the computer, such as Logic, ProTools, Garageband or in my case of Windows, I use Acoustica Mixcraft. Per what I read from Behringer, your Mixer comes with some recording software, called DAW or Digital Audio Workstation software. And you can either record a live sound coming through amplifiers in the room to microphones to the mixer to the recorder, in which case you can get all the rooms characteristics, plus sounds bleed into each other quite a bit. The second option is to get signals more directly, so for example, imagine you have 2 microphones (Shure SM58 microphones are famous) directly into the Mixer or through any effects to the mixer (if you have them) and guitar also to the Mixer either directly or through amp or effects, then monitoring the signals with your headphones and recording directly to your recorder. So it sounds like you just need more microphones, and a couple stands, and cords for all the inputs. Consider getting a "pop filter" for microphones. And getting quality microphones for live recording is it's own issue. Do some recordings and adjust levels so nothing is peaking (usually indicated by red lights which means you will get distortion) and you should get a good signal to the recorder. If your camera has a way to record sound, or through the other mic and if the sound is pretty good, then you can consider mixing the video and audio sound together for a more balanced sound, which will include the characteristics of the room.
In summary, it sounds like you need, some of which you have:
An acoustic-electric guitar, with a guitar cord/cable to the mixer, or 2 cords, one to the amp and second to the mixer.
Two microphones, two stands, two pop filters, two mic cables (you can look up the difference between XLR and TRS (1/4") plugs as you will need probably one XLR-XLR cable and one XLR-TRS (1/4") cable.)
Mixer with power cable and USB cable.
Computer with DAW software. (Audacity is ok, but you should also already have a DAW with your Mixer, per Behringer)
Additional effects for microphones and guitars per your taste (such as vocal or acoustic guitar pre-amps and effects).
You will also probably want some quality headphones. And at least one person can monitor the sound, or even a different person than those who are playing.
If you monitor your own sound or both of you, then you need to get quality headphones. I use Sennheiser HD215, which I got second hand for $75. But you'll need a way to reach the mixer's headphone jack and a way to get 2 headphones if you want it for both people. Alternatively, you should get at least a small quality amp to monitor the sound, so you both can hear yourself. That could be something like a Roland Mobile AC (very light, small and cheap), but have to adjust things so you don't get feedback. Anyway, that's all the time I have for now!!
