Overview
The actual sources themselves are variously the responsibility of the Director, in the case of actors, the Musical Director, in the case of a band, or the Scenic Sound department, in the case of playback devices.
In some cases the Live Audio and the Scenic Sound departments may be merged into one, however they are listed here separately in order to prevent confusion, as the substantial responsibilities do differ significantly.
Responsibilities
If using radio microphones, you should coordinate with the director and stage manager about who needs them and when mic changes will happen (if at all).
It is also worth talking to the costume designer about how the beltpacks will fit into the costumes, particularly in case of mic changes.
If there is a band, you must find out who is in the band, and plan how they will be mic-ed, be it section mic-ing or close mic-ing.
You must work out how speakers should be arranged and used, this is of only minimal use in the Playhouse, Pilch, and BT, all of which have good standing rigs, and more relevant in the O'Reilly which has more flexibility with speaker placement.
Ultimately it is your responsibility to mix the show. Think carefully beforehand about how to route the audio within the mixer, which will depend on the requirements of the show as well as the mixing desk available.
Members
Within the Live Audio department the role of a sound designer is to coordinate mic and speaker plots, and lay out the signal flow within the mixer.
The Mixer operates the desk during the show, for a large musical with a matinée it is usually advisable to have two people who can mix, mixing two shows in one day can be a lot.
It is also possible that particularly big shows may want two mixers concurrently on a show, it can sometimes be sensible to have two mixing desks for various reasons, either one for band and one for vocals, or one for FOH and one for monitors.
An A2 helps the A1 set things up, and then coordinates radio microphones during the show, mic-ing and de-mic-ing actors.
Resources
This page has details of every mixer you are likely to encounter in Oxford
This is an excellent text written by one of Broadway's top mixers, and the ebook is available free to download from SOLO.
This piece of software handles DCA assignments excellently for you, it supports the OP CL-3, and most desks you might hire from HTS or similar.
It is available from https://theatremix.com, and costs approximately 8USD for a one week show, though it depends on which console (it's free on the Startech M32), and there is a 2 week free trial for your first show.