Brandon Goforth
First, the best and only advice to give is to check out each state that you're going to be traveling in to find out what their laws are.
The bad thing about asking a question such as this in a non-gun forum is that you're going to get a lot of answers, and some may be from misinformed individuals.
Second, keeping a gun out on your seat is a VERY BAD idea. If a passing motorist sees it they'll probably call 911 on you. Also, if you do get pulled over (speeding, changing lanes w/ no signal, whatever) and a cop walks up and sees a gun out on the seat, you'll probably be taken out at gun point and cuffed until he/she figures out what is going on. Trying to cover it up will only make matters worse.
Getting a Concealed Weapons Permit or a Personal Protection Permit (or any variation of it) will probably be your best bet. It will show that you care about the law and you're not just some gun toting madman.
Oh, and for everyone's sake, take some classes on how to shoot. No one likes random bullets flying through the air.
You have every right to carry a handgun in your vehicle with a concealed carry permit, but at least in OK, it's not illegal to transport firearms including handguns in your vehicle without a permit...BUT you HAVE to keep the handgun in open sight, and yes, the seat is the best place for it, not hidden in the back, not under the seat, not in the glove box. It HAS to be in plain sight. If you get pulled over and you have a handgun in your vehicle you must notify the officer of this immediately, but you also have to do that even if you have a permit. I've been planning on getting a concealed carry permit myself for quite some time now and just haven't taken the time to do it, but I know a lot of people that have them, and this is some of the first things they talk to you about when you take the class. With a handgun, it must be in complete open view and unconcealed, and separate from the ammo. It's the law.