While I feel that, for the most part,
ToasterLeavings' advice is indeed valid, and even rather good, as advice goes, I have to disagree on a few points.
I train to failure, especially on the last set of any given exercise. It's been my philosophy that if you're not "leaving it all out on the field," as we Ultimate players put it, you're doing yourself an injustice. Now, maybe if you're just starting out, you want to stick to what you're sure you can do for a while train to success, not failure and all that. But half the fun of lifting for me is not quite being sure if I can squeeze out one more rep, trying it, and finally finishing that last rep after a massive struggle. You against gravity, baby. Note that I am not advocating the use of improper form just to get a final rep in - training with improper form invariably does more harm than good. If you can't finish the rep, don't - there's no shame in stopping halfway, if you're completely pumped out. I, personally, like to hold the weight up as long as I can, even if I can't make it go any further, just to feel that sweet, sweet burning of lactic acid. Once it starts to hurt, rather than just burn, I let the weight slowly back down, and I am finished with that set. Note that pain is not good. Burning is good, but if you feel any sort of sharp pain, or anything that's more than vaguely localized, put the weight down and figure out what's wrong!
I like to keep a weightlifting journal. It says things like "Oct. 23: Leg press. 280: 10 reps, 260: 10 reps, 260: 8.5 reps. Leg extension..." and so on. It's a neat way to keep track of your progress, and it definitely helps you make sure you get in all the muscle groups you're aiming for. I'm going to go out on a limb here and suggest a pretty basic training schedule that's done very well for me. This one assumes you have time to work out 6 days a week... I'll tell you how to shuffle it around into 3 days in a minute.
Monday, Wednesday, and Friday are lower body days. Warm up with a brief jog (you don't want to tire out your legs completely before you start lifting; I usually make this a 2-3 mile run) and stretch. Then get a thorough lower body workout in - you want to do squats, leg extensions, calf raises, all those lovely exercises that make your legs strong. Cool down and stretch. Stretching is important, 'cause it helps prevent delayed onset muscle soreness, and also because as your muscles get stronger, they will tend to tighten up.
Tuesday, Thursday, Saturday are upper body days. I usually do a more strenuous warmup run on these days - 5-6 miles at an 8-minute pace is about right for me. This gets the blood flowing, helps work out the soreness from the previous day's lower body workout, and improves your aerobic condition. Of course you should scale the warmups to what suits your current level of conditioning... if 10 miles is a warmup to you, then go for it (but you probably didn't need my advice anyway); if walking a mile leaves you gasping for air, give it 10 minutes on the treadmill. The point is to get the blood flowing, get the muscles warm. Stretch, then go for a variety of shoulder, chest, arm, abdominal, and lower back exercises (if you prefer, you can work abs and low back on your lower body days). Cool down and stretch again.
Sundays I like to go for a nice long run... this is optional, but it helps me get rid of soreness, and I just like to run.
If you can only get 3 gym days a week, for whatever reason, shorten the warmup run to about 10 minutes, and work both upper and lower body. Make sure to go for a jog or something on your non gym days.
Weight training has been really rewarding for me... recently I've cut down on my lifting time to work on plyometrics and other forms of exercise that really don't become options until you're in good enough shape to do them correctly, but I still try to get down to the weight room a couple of times a week to keep up my basic strength. I enjoy lifting, and I believe most people can too, once they get the hang of it.