What are some good gym workouts to improve rock climbing endurance and strength?

The best way to get rock climbing endurance is to climb more. Climb longer/taller routes. Your arm muscles (the "Popeye muscle" on your forearm) will come with time and practice. You can also get a rubber ring to squeeze (http://www.backcountry.com/outdoorgear/Black-Diamond-Forearm-Trainer/BLD0527M.html?avad=31835_ad2bf5d) to build up forearm muscles. See http://www.indoorclimbing.com/Forearm_Exercise.html for more potential exercises you can do. And everyone is right - you are not using your legs enough. The most common mistake (especially for men with good upper-body strength) is to reach far and do pull-ups instead of moving your legs higher when they start climbing. Try to use your arms only for stability, and get your legs as high as you can (frog-like position) before you move. Keep doing small steps up with your legs, and you'll be able to climb better Don't worry, it all comes with time and practice.

If you just started climbing, odds are you are also "over-gripping." Part of climbing is to hold on to the holds with as little exertion as possible, without falling of course. If bouldering, one of the simplest things to do is count your moves. Find a number where you tire but are not exhausted. (This number is similar to when you work out with weights in the gym.) Then do 3-5 sets with 1 minute of rest in between. Over time, increase the number of moves. Eventually you'll work your way up to 20, 30, 40, 50 moves before taking a rest. If top roping, you can do something similar. Find a easy to moderate route and climb it, up and down and then repeat at least once. Then your partner can do the same. Then move to another wall and do the same thing. You can also add forearm exercises to your "arms" workout at the gym. Do wrist curls and extensions.

For endurance training you can pick a route a grade or two below what you climb at your limit. Climb it up and then down climb it, repeat a few times and stop before your arms get pumped. Take a break then and repeat (perhaps on a different route for some variety). Use these easier routes to pay attention to your breathing as well as other factors mentioned above (how tight are gripping the holds, how soft/hard are you placing your feet on the holds, etc). This has worked for me very well to recover endurance after long periods (months) of bouldering only sessions, though climbing an easier route repeatedly can get tedious at times. For strength training, bouldering is great. Pick problems at or slightly above what you climb usually, if you fall on a move stop and take a break, then attempt the problem again. It helps also if you have any particular climbing goals as this would help you plan your training better. If you are just beginning though (as your post suggests), you are better off climbing as much as you can at first so that you develop a feel for the movement on the rock (or holds at the gym) and then later on devote time to more climbing specific training. At this point, you may want to check out Eric Horst's "Training for Climbing", it's a great resource.

Leg muscles are much stronger and have more endurance than arm muscles. You were likely beat after your climb because you used your arms too much (which all my rock climbing instructors have told me is a common mistake for men). Rather than attempt to strengthen your arms, try to correct your technique so that you rely more on your lower body for the climb. Develop a habit of using your legs instead of your arms to lift yourself.

Of course, there's a variety of exercises to strengthen your arms and hands and fingers, that everyone else has alluded to. However, don't overlook abs. You may not notice your abs being tired after climbing, but with stronger abs, you are better able to control your position and balance, leading to needing less arm strength/endurance.