Some interesting advice here and just to be awkward here's mine. Starts with some basic thoughts and finishies with tips.
I agree with Sure that experimenting on softer surfaces is safer. There is an issue with this however and being a cross country runner who moved to roads I can safely say the two are very different and what works on one needs to be adapted for the other.
Gait analysis is very valuable and getting the right shoes is crucial. I found the brand that work for me and stick to them religiously.
Correct gait is to strike slightly to the outside of the heel, roll along the outer edge of the foot, then cross the ball of your foot and take off with the big toe. Seeing ff you run or walk with good gait needs only a look at some well worn shoes, ideally a set you have only used for either running or walking, not both. Wear would be mostly at the outer heel and big toe, but not too steep at the heel meaning you are striking the heel with your toe to elevated, the rest of the main wear should show the rest of the path of your weight along your feet. This doesn't replace a proper treadmill analysis, but i had one where the treadmill was too narrow so I had to holg my arms up and was running un-naturally so got false readings.
Running is everything regarding bio-mechanics. I was a good level distance runner and the part I got wrong was my arms crossing my body. Correcting this cut over 2 minutes off my half marathon time in a year, improvement I didn't expect especially when it added over 5 to it in the early stages.
There are people who have done very well with poor running styles, my most loathed is the nodding dog Paula Radcliffe who is so proud of what she did despite ignoring coaches, unfortunately not smart enough to imagine what she could have done if she hadn't ignored them.
Lifting feet off the floor. The worst thing I hear people say about running 'get those knees up.' Running is a high impact activity, there are complex calculations regardign the impact based on bodyweight, speed and the distance you land from. Cross country involves bringing your feet above obstacles, the road or pavement have few of these, as such my feet are rarely more than a few inches from the ground, why waste energy and increase impact un-necessarily. I am not running upwards, so only forward energy is worthwhile and I am now almost 1.5 times my ideal running weight so reducing impact is all the more important.
Things to remember starting at the top.
Head shouldn't be up and down all over the place, because your running should be smooth. Face relaxed head straight allowing easy breathing.
Shoulders, relaxed unless sprinting. There are a number of people who run with shoulders lifted, creating massive tension and risking RSI style neck injuries. Smooth style means being relaxed, not easy when focussing hard.
Arms. Relaxed and moving forward and backward if going straight and flat, slightly up if going up hill, down on downhill, more swing from the right when going left etc. Arms direct you a lot and crossing the body when only one foot is in contact with the ground at a time generates twist on the knees.
Torso. Correct alignment and relaxed, allow breathing to happen without affecting posture. The torso holds everything together, keep it under control, but not tense.
Legs. Send them where you are going, if you are on a steep up hill lift them, flat, send them forward, seems idiotically simple, but it's where many go wrong. Make sure your knees go in parellel, many run with knees buckling inward or outward then declare running is bad for the knees. Running badly is bad for most things running well is one of the only exercises our whole bodies were designed for.
Feet. Strike pattern was covered, keep them in a nice alignment again and take care of them. The get the most battering so be nice.