Each element trains a different skill.
Shadowboxing is great to develop your posture, structure and accuracy, but you will never develop significant power using this method alone. In fact, full power air striking is bad for your joints, so don't do it. Use shadowboxing to develop proper form and above all, visualize the opponent.
Partner work is so diverse that very few generalizations can be made. You want to escalate from static movements to improvised, hard contact training if possible. You also want a diverse range of body types and skills. What you learn will depend on what the other person brings. You might be partnered with a smaller, more timid person; that's the time to concentrate on form and sensitivity. With a bigger, more athletic guy, you want to work on timing, endurance, speed and the "hard" elements of training. You shouldn't neglect anything else, but you should rise to the challenge. One of the worst things you can do is blame the other person for using "too much strength" or similar garbage. Let them work on their end and you work on yours. You meet in the middle with technique, not words.
Oh, and make sure you have at least a mouth guard for any heavy contact training.
Work against targets for striking power, naturally, starting with static exercises (working a punch over and over again), and then up to what is really partner work with pads instead of a warm body to work with.
Contrary to what folks are often told, you should *always* rise to accept things that come up out of the normal order of training, such as sparring without "working up" into it with intermediary exercises. You should put yourself in situations where you don't feel prepared and which break your normal routine. The only situations that are exceptions is where you are worried a known physical limitation will create a problem. Don't try something where you'd probably hurt yourself, but do try something where the risk of injury is slight, but your level of mental preparedness is low.
For the last, I'll give you examples of a good challenge and a bad challenge:
Good: At an aikijujutsu class, a BB asked if I wanted to work striking combinations with him. I said yes and was nicely schooled by him, but tward the end I rose to the occasion. I trusted him not to seriously injure me (and if you don't have that level of trust in a club, leave!).
Bad: At a karate school, I was asked to squat under kicks thrown by my partner. My partner was 8" shorter than me and could only kick to waist height. I asked to modify the exercise, s it was obviously impossible for me to do this with proper form. The brown running training shouted at me. I did it anyway. I missed 2 days of work because having to squat and then bend my back since squatting below 90 degrees wasn't good enough.