Even if I would not recommend to everybody to work specifically on their pushups, they are a great upper body workout that can be done anywhere and allow for countless modifications to suit everyone.
But whatever the pushup you want to do, the first thing you have to master, is the PLANK.
You need to have your abs bracing and holding your lower back in place so when you will move your chest, your butt will follow at the exact same time. To do so, you'll need to tuck the tailbone in, (no popping the butt up) and suck the abs in.
You can train at planking by starting on a kitchen bench or diner table as seen here:
Note that you can do ALL the pushups on the table instead of doing it on the floor. Lets call it your level 1.
You should always start at this level, and wait the next day to see where your body hurts before attempting a progression. And if you can't do 20 pushups without swinging the hips: you'll have to train at this level until you master it.
Now that we have the body position sorted, what about the hand's position?
There are a lot of positions to do pushups, and as long as your elbows are not in front of your shoulders, you'll be right.
I see too often people sliding their body backward to slide the weight of their butt on their knees instead of their arms, hence injuring their shoulders by ending up in the wrong position... Remember, you should be on a plank, hip inline between your knees and shoulders.
Watch this video to see what I am talking about.
Finally, how to give you the motivation to exercise?
Do it in music!
Get yourself set, and follow me on this track.
Adapt your level and your hand position to a comfortable one.
Did you get to the end?
Well done, you just did 50 pushups!