Remove your current mixer from your mixing desk. Disconnect the inputs and outputs, and set your mixer aside.
Connect your new 16-channel mixer to your recording device. Connect the outputs of each mixer channel to the corresponding input on your recording device. Make sure you pay attention to matching channels, so you don't run into problems later, such as recording a channel to the wrong track.
Set up your input system. If you have a one-room studio, this is as simple as connecting each microphone/sound source to the desired channel input on your mixer. If your control room is separate from your recording room, a good way to do this is to connect a 16-channel snake to your mixer, and place the input box in the recording room (particularly if your studio doesn't have in-wall input wiring). Again, make sure that you pay attention to each input when connecting your snake, so you don't run into recording problems later.
Connect your microphones/sound sources, and begin recording.