Before climbing in a new gym you will have to sign a waiver, and if you are under 18 your parent will have to sign the waiver. Most gyms have their waiver online so you can download it and fill it out before you arrive to save time.
Call the gym ahead of time to make sure they will be open to the public for climbing. Throughout the year gyms will sponsor climbing and bouldering competitions and may not be open for climbing during those times. The last thing you want to do is drive 3-4 hours only to find out you can't climb where you get there.
If you have never climbed before you will be required to take a belay lesson. If you are an experienced climber most gyms will allow you take a test to show you can belay and you can avoid the lesson. NOTE: Be aware, some gyms will still charge you a nominal fee for the belay test.
Make sure you are aware of and understand all the gym rules before climbing and abide by them. It is their facility and the rules they have in place are for your safety.
Some outdoor areas will also require a waiver to be signed and checking online is a good way to find this out and possibly have the waiver filled out ahead of time. Also, most areas I've climbed at, you have to check in at the ranger station (state parks) or sign another type of log book posted near the trailhead when you enter and sign out when you leave. This is so if you get injured or go missing someone has an idea of where to start looking for you.
Now for my personal rant on outdoor climbing - LNT - LEAVE NO TRACE! From talking with landowners, and leaders of various groups such as The Access Fund and the SCC most climbing areas get closed down because the climbers (that's us!) don't respect the land and the privlege we have to climb there. Every single time we go out climbing we pack out more trash than we brought in with us and it is frustrating. We are all adults and should be able to pick up after ourselves; Throw your cliff bar wrappers and empty gatorade bottles in your pack and when you get back up to the top of the trailhead throw them in the trash.
Also, stay on the already established trails in and out to the cliff face. Yes, the trail winds back and forth some and usually doesn't follow the most direct route in, but if you're going to spend 3, 4, even 8 hours climbing you should be in good enough shape to be able to walk the extra 50-100 steps in. This helps preserve as much of the natural beauty of the area as possible. Again, leave no trace that you were ever theere!
Check the weather - If this is a day trip that may be as simple as waking up and looking out your window. However, if you're planning a 10 day trip 2 months from now you can go back and chekc the weather for the last 10 years. If historically it has rained 7 days out of 10 for the last 8 years then I'd look for a different week. It isn't fun sitting in a tent all week while it rains on you.
Again, depending on where you're going and for how long, make sure you pack properly. A one day sport climbing trip all you may need is your harness, shoes, chalkbag, one rope, and a dozen quickdraws (although I'd still recommend a first aide kit). A longer trip I would recommend two ropes (We had a trip recently where we found the outer sheath of our primary rope severely worn in one spot). Although it was probably still safe to climb on it we didn't have to worry about it; We left it in the tent and climbed on out other rope the rest of the trip. Also, unless you want to be making trips to town make sure you pack enough food, drinks, toilet items, matches or a lighter to start camp fires, cooking and eating utensils, changes of clothes, etc. A good first aide kit (with the number of the nearest hospital and directions), spare pair of shoe laces for your climbing shoes, extra quick draws in case you drop one and have to retire it, and things of that nature.
A good rule of thumb when packing for a trip is "Hope for the best, but be prepared for the worst."
Strength - Good for bouldering, and short, explosive moves on the rock.
Endurance - Good for sport climbing, trad climbing, longer multi-pitch climbs.
Rest - Vital for allowing your body to recuperate. I try to never climb on back to back days and if I do I always climb easy one day.
Focus on going up the wall.
If you want to have a long and successful climbing career then warming up is vital. When I was younger I could walk into the gym, pull on my shoes and start cranking routes. I can't do that anymore and know that I shouldn't have done it then. Get to the gym and spend 15-30 minutes loosening up. Stretch your calves and thighs for good high steps, your back, your shoulders and arms, and work your wrists and fingers as well. Then when you get on the rock start with a couple of easy routes to get your body prepared for climbing. After this you can pull hard and not have to worry as much about tweaking a muscle or tendon. Finally, take a few minutes after you are done packing up your gear for the night to again stretch your muscles for just a few minutes before going home. Your body will thank you for it in the long run.