Check out this article written by Seth Lytton. It provides some good ideas for how to properly warm-up before climbing. Most importantly, it mentions the importance of mimicking the more aggressive moves on your project when you're warming-up. In other words, if you know there's a massive drop knee in the middle of the crux, you should be practicing drop knees on your warm-ups.
I will say one thing: avoid dead-hanging on holds to warm-up your shoulders. Shoulders are inherently lax joints and the last thing they need is for you to hang on your arms loosely. Seth doesn't recommend doing this - he simply says to hang on good holds. The concern is that this encourages disengagement of your shoulders. You can prevent this from happening by keeping your elbows turned in while hanging or doing pull-ups. Check out my video bellow to help you understand. It's a repost but hopefully you'll get the idea.
By William Bateman