Push ups are a real pet peeve of mine. You think, hey, it's just down and up! Yay, pecs! Then I see all kinds of worms and back cranking and shoulder rolling happening. It's really a lot harder than you think it is and the push up deserves more attention and respect.
1) Prioritize the spine
Like you've never heard this or some version of it. It's one thing when you can't do a full push up to let yourself "snake" back up to the start, but for godsake, man, don't make it look like you're trying to hump the ground or make out with it when you're descending!
I see this a lot. People will reach their face and/ or their hips towards the floor to shorten the distance their arms have to take them. You're not fooling anyone, and you're spine isn't going to be too happy about that in the long run. Don't get weird with the floor, just bend the elbows and do what you can.
2) Use your back!
Most think of this as a chest and triceps exercise. But, just as in the bench press, if you aren't engaging your lats and back then you aren't doing this right.
At the top of the push up, many coaches say you should have fully protracted shoulders and scapula. That's all fine and dandy, but as soon as you start to descend, you need to pull your shoulders back in line and get lat engagement and external rotation through the rest of the movement. Those muscle are much bigger and will support you through many reps more than the smaller muscles of your anterior delts alone.
3) Vertical forearm
This one should fall in place pretty readily if you follow the previous point. Again, much like the bench press and shoulder press, your forearm should be inline with the direction of force. You're pushing into the ground, so your forearm should be directly perpendicular to the floor.
Another way I've heard it described is that the elbow is like the knee and the shoulder is like the hip. Just as you don't want to shoot your knees forward in the squat, you don't want to shoot your elbows back. (Let's not get into the arguement about knee, shin angle, and torso angle here. That's why I don't use that description, personally.)
Okay, so in the end: (1) Don't get weird with the floor. (2) Put some back into it. (3) Elbows up!