AARF! - Approach, armswing, release, followthrough tips
Developing a consist approach, armswing, release, and followthrough can be the quickest way for a new bowler to greatly improve their game and have it reflected in their scores. The two most commonly accepted approaches are the 4 step approach and the 5 step approach. I personally favor the 5 step approach, but ultimately whatever you are most comfortable with is best. The last 4 steps of your approach with a 5 step approach, or your first step in the 4 step approach are the 4 vital steps. The first step for a right handed bowler is with the right foot and you push the ball forward at the same time.  On your second step the ball will fall and be approximately at the 6:00 position. On your 3rd step you will start to bend down to get closer to the lanes and the ball goes behind you to the 9:00 postion. You push into your slide, or 4th step, and the ball comes forward to be released onto the lanes just behind your ankle.
Your armswing should be free, loose, and unencumbered like a pendulum. Ideally you want to be able to push the ball slightly upward and forward and then let it free fall down, behind you, and then back forward to the release point with no help or force from your arm at all. A good tip I got one time was to imagine there is a thumb tack sticking out of your bowling ball into the palm of your hand as you push it away so as it falls you have to move your hand quick enough to keep from being stuck by the thumb tack.
Your release should be even with, or slightly behind, your ankle. Your thumb should come out of the ball first and then you project the ball onto the lane and lift with your fingers to get rotation on the ball. I personally find it most comfortable to project (lead/guide/aim) the ball onto the lanes with my middle finger, however that is personal preference.
Your followthrough should be straight forward towards your target and not to either side. After extending to project the ball down onto the lanes the arm should come up with the thumb pointing up and the palm facing the lanes as if you are getting ready to shake hands with someone.
This is a basic concept explaining approach and followthrough; However, your best best is to consult with a USBC certified coach in your local area.
It has been my personal experience, and in my opinion, that a strong mental game is much more important that a strong physical game. The physical aspect is fairly straightforward once you learn the basic mechanics of approach, armswing, release, followthrough. However, holding yourself together mentally and being able to calmly analyze your game when you are throwing good shots and not striking can be a much bigger challenge, especially if your opponent is striking no matter what they throw down the lanes.
Always remember to focus on positive thoughts and keep negative thoughts out of your head. If you throw a bad shot, or even worse, a good shot and get a bad break, let it go. Realize that everyone gets bad breaks and move forward to making the next shot a good shot.
If you want to have a long and successful bowling career then warming up is vital. When I was younger I could walk into the alley, pull a 16lb ball out of my bag and start throwing. Now I can't do that anymore and know that I shouldn't have done it then. Get to the lanes 15-30 minutes early and take the time to loosen up and stretch your calves and thighs for a good deep knee bend and release, your back, your shoulders and arms, especially your throwing arm, and work your wrist and fingers as well. Then when the lights come on for practice throw the first few balls a little easier to get the blood flowing before you start firing those hard strike shots down the lanes. Finally, take a few minutes after you are done packing up your bag for the night to again stretch your muscles for just a few minutes before going home. Your body will thank you for it in the long run.