Best bet is to check with the states that you will be travelling through - many of them should have this information available on a state police (or similar) website. If they don't, they should have a point of contact who can provide that information. If you don't want to do this, then your best bet is to have it in a case in your trunk, unloaded of course.
As for notifying any officer you may come in contact with about transporting firearms - that is really a toss up. I have spoken with many officers through the years, as I often found myself in the same situation, and it has been about an even split when they were asked if they wanted to know about any firearms up front or not. If any actions you take will put you in proximity to the weapon, definitely let the officer know beforehand, as you wouldn't want to put yourself in danger. Likewise, if anything the officer does will put him in contact with the weapon, let him know so it doesn't appear that you are hiding something. And as was already stated - if asked specifically about any weapons, do not lie.
Edit: Oh yeah, if you have a CCW permit and wish to carry on your person, check with your state of issue to see if they have any reciprocity agreements with the states that you will be travelling through. You may be able to carry legally, but you will still have to abide by the carry laws of each respective state.