The basics of grip and set up create the best start to learning to swing the club. Grip is the only attachment to the club and set up is impact, both need to be correct before any movement in the swing occurs.
The "open tube of toothpaste" is a great visual and a usable refernce on the hand pressure, thanks!
Yes, your grip is your hold on the golf club. Many people have the same issue with holding on too tightly. You would want to try to grasp the club as if you had an open tube of toothpaste. Be careful to not gush any toothpaste out of the tube. Too tight of a grip will also not allow you to keep the tension out of your arms and shoulders which will definately make it harder to swing freely.
Does holding on for dear life qualify as a "grip"? Seriously, I find that the more tense I am the tighter I hold on - and it seems to travel up to my elbows (which get stiff and tend to lock) and on to my shoulders which begin to tense up around my ears.