I started a new program a couple of weeks ago and my upper torso (pretty much all of it) is really sore. Any tips on how to lessen that between workouts?

Well, from the outset, such reaction is pretty normal and in fact should be expected anytime when You start new type of exercises/routines etc. It will go away, eventually, as Your body will learn to adapt to the new workout. Meanwhile, You can try any or all of the following  - as every person is unique, it is not guaranteed that one particular method or combination will help, anyways.
To prevent or lessen soreness, before it happens - stretch the areas, which You are going to work out.
If soreness has already happened You can gently massage sore muscle fibers / take a low dose of ibuprufen / warm up in bath or sauna and massage them there.
You can also google around and search for anything related to DOMS - delayed onset muscle soreness.

Thanks

Epsom salt baths really help.  Epsom Salts are an anti-inflammatory so they really relax your muscles.

Quote from: Liam Erik on July 15, 2013, 10:42:32 PM
I go back and work out again the next day when I'm muscle sore as heck.  It seems counter-intuitive but it works... you will feel much better afterward.  If you try babying it, sometimes you can get sore spots that stick around for weeks.

Also, when you are just starting, there is another, worse pain for a while until you get used to it, so maybe you are experiencing that.  Thankfully that one goes away.

Unfortunately that isn't an option for me, I'm on a high intensity program and it has mandatory rest days.
I wanted some ideas that could tide me over until I was allowed to work the muscles again. Thanks though.

Ibuprofen or Aleve are your best bets.  Heating pad before you work out to warm the muscles up AND stretch before and after your workout.  Ice the muscles at night before bed.

In my Aerial silk classes, one can well imagine we get sore quite often. A lot of us use these things called rollers before and after we stretch to roll all of the lactic acid out of our muscles. They hurt like heck when you're using them but afterwards you feel ten times better. You're supposed to find the place where it hurts the most and put pressure on it until the pain is diluted by half, then roll the area out. It's always worked quite well in my experience.  If you search "workout roller" in google it should come up with a bunch of them.