Here's what I can tell you, Ice or a cold bath is used mainly to stop swelling/inflammation, so it shouldn't help much with soreness, and a hot bath or warm patches/ointments help circulate blood, neither are really meant to help soreness. If you are extremely sore, try some NSAIDs like Ibuprofen or Advil. Either way, you should not get hung up on healing the soreness, because as it has been mentioned earlier, it is really only noticeable when you first start out. Squats will keep you sore after the first week of doing them, but you get used to them. Once you get consistent enough and really get into a groove you aren't even going to be sore the next day unless you go with an extremely strenuous workout, and that will probably only last you a day. So there is no way to stop "heal" the soreness, you can only numb the pain, but your body is still going to need time to recover, just don't stress over it cause it won't even be an issue a month into training.
If you're sore more than a week with rest and all that, probably not sore from the workout, you probably did strained a muscle, and NSAIDs will help with that, but you should probably go to the doctor. I strained my bicep my second week into training and couldn't straighten it out for two weeks before going to the doctor and getting a NSAID to clear it up in a week.