Connect the microphones into the mixer and run the mixer output cable into the computer. This will be the circuit for the recording session--audio going into the microphone gets transferred to the mixer, which manipulates it i(volume, levels, etc). It then is sent to the computer to be digitized and edited.
Connect a pair of headphones into the mixer's "out" port. Place them on your ears. This will give instant feedback for the audio.
Launch the computer's audio recording/editing program and create a new session.
Sing or play into the microphone and listen to the sound. Determine if it the sound you wish. If not, try adjusting some of the "faders" on the mixer to raise or lower the volume. Perhaps the treble or the bass should be adjusted on the mix. This is all handled through specifically indicated knobs on the mixer.
Record a short take of audio, performing into the microphone and being cognizant of the headphone feedback. Listen through the recorded audio and assess the quality again. Does it need to sound bigger? Try including "reverb" on the audio which is available on some mixers and most audio programs. Reverb is a slight echo and delay in sound that makes an instrument sound "bigger," as though it was recorded in a large hall.
Hook up a second microphone to the mixer rig and bring in another instrument. Recording two or more instruments on individual microphones is known as multitrack recording. This allows each instrument to have a dedicated microphone and individualized audio setting. The two microphones will have two separate tracks in the audio program and can be edited independently.