Recent AMA Shannon Slaton has a great book on the subject of mixing,
Line by line mixing using faders is standard practice. Someone walks on, fader goes up, they finish their line, fader goes down. That being said, in the digital world, most people mix using scenes and DCA’s, so each scene or chunk has its own saved scene/snippet that puts the people and groups you want on DCA faders at your fingertips rather than using the faders for the actual channel. That way you’re not hunting through a whole desk, or various layers to find everything.
That's a big question requiring a long answer. Some might say a book length answer:
A lot of skills will transfer over, but there are some very specific differences that you'll encounter:
1) Vocals, vocals, vocals - It's all about the vocals. Even more than a rock show, you gotta keep those vocals on top.
2) Compression - You'll probably be using less compression in general than on a live concert gig. Most musical theatre shows really want a decent amount of dynamic range. There's a lot of expression to be had in a quiet moment that has room to breathe. Some more modern musicals have a more rock feel and might benefit from that more compressed kind of sound, but any show written before the mid-2000's is gonna want to breathe.
3) Monitors - If you can, avoid folding back the actors. They need to hear the orchestra pit for tempo and timing (talk to the conductor to find out where in the pit the "clock" for that particular show lives, and make sure that it is audible everywhere on the stage.) You'll likely need every single dB of gain-before-feedback for your actors. Don't EVER start a tech rehearsal process with vocals in the monitors. You'll never be able to give them enough and you'll regret giving them that option.
4) Mic placement - Learn the holy trinity of mic placements (wig/forehead mount, temple mount, and earworn/earset.) Please don't just tape a lavaliere to their cheek, it looks awful and doesn't sound that great either.
5) Mixing - Musical theatre mixing is full-combat audio from the very first downbeat until the end of the final bows. It's intense, and amazingly rewarding. Look up "line-by-line mixing" or check out this post from Soundgirls, this summary of musical theatre mixing goals, and this book by Shannon Slaton which is an excellent entry into the art and practice of musical theatre audio from prepping a show through opening night. But to summarize, you have a bunch of walking omni mics in close proximity to each other belting out sound at the top of their lungs. Recipe for a comb-filtering nightmare. Solution? Keep as few mics open at a time as possible. Literally mix the show line by line.
6) DCAs and Snapshots -- They are your best friends. Coming from someone who cut his teeth on line-by-line mixing on an analog board with no automation, recallable DCAs are the greatest gift to musical theatre ever bestowed upon us by the gods. Learn them, use them, love them. Become an expert in how your console handles / safes snapshot recall.
7) Paperwork - there's a shit-load of it and it is all necessary and useful. Mark up that script and keep it where you can read it. You'll be following it very closely until you get the show under your fingers. Mic plots, DCA plots, patch plots, pit diagrams, learn to love your paperwork.
8) Other sound department responsibilities -- It sounds like your company might already have a designer or tech on staff to handle this part of the job, but smaller companies who don't have a systems tech might assume the FOH guy is also handling stuff like production comms and sometimes conductor cam / video monitoring falls under sound's umbrella too. So make sure you know the extent of their expectations.
9) Qlab - is pretty intuitive. If you know your way around a modern console you'll understand the basic workflow and concepts (routing, matrix crosspoints, etc). Between you and your designer you'll have to work out who is doing the actual cue programming. If you have never used it before, Figure53 has some great basic tutorials. And there's always /r/qlab if you need help. (qLab can recall console snapshots for you over midi which is nice when you only need one <GO> button. Conversely, you can have your console snapshot recall trigger qLab playback if you'd prefer to use the console's recall button as your <GO> instead.)
10) Stage Managers - that'll be the last major difference between concert mixing and musical theatre. It differs slightly from company to company, but you will likely be taking cues from a stage manager regarding the timing of SFX playback. Often that's through a cue light, but the actual method of cue delivery will depend on your stage manager and your venue.
Musical theatre mixing is my absolute favorite kind of audio work. It's INTENSE but extremely fun and rewarding. Enjoy!