This is something that you just get used to.... and the reason I tend to buy cheapo screen protectors in packs of five or more! Make sure you have a suitable cloth and preferably a small card to push bubbles (credit card will do but they're a bit big and a little stiff).
1st step, clean your screen. Any dust will lead to bubbles under the protector which will annoy you. Get rid of any finger smears as well, or they'll be there until next time you change protectors. The best thing to use is a soft dry cloth and patience. No scrubbing!
Then just lay the screen protector on top of the screen and see how accurate you're going to have to be when sticking it on. If it's cut exactly to size, you'll need to be totally precise (which I can only do when working from one side to the other). If it's cut a bit smaller, you have a better chance of getting the thing on without any sides lifting and causing bubbles you'll never shift.
When you've worked out how you're going to position the protector, take off the film that is on the side of the protector that will sit on your phone. There's usually a tag with a helpful number 1 on it to help you get the right one.
Now, hold only the very edges of the protector and line it up with the phone screen. Your aim is to use the flex of the protector to smoothly "roll" down or along the screen in a way which will trap as few bubbles as possible. Once you're lined up, go for it. I tend to put about half an inch on the phone and then adjust the line... any more and it's stuck too firm to shift. At some point you'll just have to keep on going, roll the protector on the phone and follow it up with your card just behind the bend.
Hopefully you'll have all the edges in the right place. If they overlap the edge of the screen then you may as well take the protector off now and start again. If you've got it on straight you can take off the top layer from the protector now. I find that usually this will lift the protector up a bit, so pull directly downwards so there's very little lifting force and not enough of the protector will lift to allow it to shift.
Now use your card to chase the bubbles to the edge of the screen, always taking the shortest route. A thumbnail will also work for this but a totally flat card will tend not to split the bubbles into smaller bubbles which a curved nail will do. Just don't go mad and scratch the screen protector. Do NOT be tempted to prick the bubbles.
If you've done all this correctly, you'll have a screen protector that you can hardly see and no bubbles. If you've left any dust on the phone, you'll now realise that you can't push dust to the edge and you've got a permanent bubble. If you can live with it, it's up to you!
Sorry for the essay - but can you tell who puts screen protectors on all the new phones at work?

I won't let users pick them up until there's a protector on there.....