In reply to chrisa87:
First and foremost this will improve with experience, the more time you spend doing something the more relaxed you get about doing it. It's not something you really want or need to rush and with it comes obvious the danger of complacency.
That said, you mention you're gripping too hard which is a common problem. It really helps to know how little grip is needed to stay in contact, that way when you identify the problem you can relax in relative confidence and shake out. It sounds easy but even experienced climbers regularly get it wrong.
Over-gripping can also be a symptom of poor body and foot positioning. Getting your feet right and your weight over them can really unload your hands. With enough body tension and the right footholds you can basically get a hands off rest on even vertical ground.
The exercise I find useful for this is traversing but working in some ups and downs on reasonably good holds, nothing too steep. It makes for a good warm-up too. As I move I focus on shifting my weight around and sometimes grip around to unload the limb I want to move. Moving slowly and methodically, always in control even for moves that require a little momentum, helps ingrain this. Aim to place your hands and feet right first time, it won't always happen and often as your weight subsequently shifts so will your contact patches but it's a good aim. All the while you're trying to stay relaxed, to avoid getting too pumped. Relax your grip, shake out between moves and even mid move once you've achieved the slow steady progress you're aiming for. If you misjudge it and slip off then there's no harm done, it's not a route failed or a proper fall.
The next problem comes in applying this calm fluid style to more stressful positions, on lead, pulling up slack to clip etc. Again there's no substitute for experience but building trust in your kit and most importantly partner by experiencing some controlled falls.
jk