Hi PPL152,
There's lots of good advice here. You're on the right path with your practice, keep with it, and don't be put off. Good judgement comes from experience - experience comes from bad judgement.
The problems which you are having are perhaps a bit more noticeable on the C172, but pretty common to most low wing loading nosewheel types. You've got too much speed at touchdown. If you're not hearing the stall horn peeping at touchdown, or imediately after, you're too fast. Fly the plane by feel, not fixated on the airspeed. This skill will prove very well worth the practice when an instructor simulates a complete electrical failure during night flying training, and you have to land with no instrument lights (or flaps) at all. (It happens for real too!)
But back to your challenge... Try these: Lots of soft field takeoffs - even if the field is not soft. It'll get you used to the proper nose high attitude on the ground, and the lower speed getting airborne. You can feel the nose wheel come off the ground (in the pedals), once it is just off, hold that attitude, and memorize it. Don't raise the nose anymore, because banging the tail tiedown ring on the ground is a very real possibility. It's bad for the plane, and your pride - not to mention your wallet if you hit it more than a scuff. Also, Practice (at a safe altitude) very slow flight with full flaps. The C172 has excellent low speed control, get used to making the best of it. If it stalls, don;t worry about it, but don;t let it spin, the recovery is hard on the flaps at 40.
When it comes to landing, assure that every landing you do (forever) follows a stabilized approach which has you crossing the numbers at the correct height above the ground and airspeed. Not 10 feet too high and 10 knots too fast. Once you have this under control, you have assured yourself that the aircraft is going to consume about the amount of runway that Cessna's flight manual says it will. So what's your hurry to be on the ground? You want to be flying don't you? If you decelerate in the air, rather than on the ground, you reduce tire and brake wear, and assure no bounces. If the plane touches down at the lowest possible airspeed, it is about to stall. This is what you want, because it can't fly anymore! If it cannot fly, it cannot bounce! If during this beautiful slow flight mere feet above the runway, you feel that you're about to stall, and drop, do not lower the nose, just add a bit of power. If you've reached that attitude, there's a good chance that you're going to have to carry that power setting to touchdown, and maybe even add a bit more. Within reason, this is just fine, but you're at the limit of a safe landing at this point. After you touch, continue to hold the nose up. it helps slow you down faster, and prevents nosegear shimmy and damage. Don't just let go of the controls once you're down - it makes it look like you suddenly lost interest in flying!
Avoid this kind of flying with anyone in the back seat for the first while. You'll get the feel much better if you're more forward C of G. If the trim is all the way nose up...okay. You're willing to put a bit of muscle into the flare right? With weight in the back, striking the tail on the ground is quite easy, and the "feel" is harder to detect.
In a C172, there is NEVER any need to push on the controls during the landing. If the aircraft is that far out of trim, there is no way you could have flown an approach at all. You will never be pushing to recover a stall, 'cause you'll be dropping too fast already. For the new pilot, short runways can get to look even shorter half way through a landing. Did you check the runway length in the flight manual? Enough? then it will be fine - don't start worrying over the numbers, it's a bit late then!
I once landed into a 1200' runway in a C182, an hour after a guy wrecked a C172 on the same runway (only hurt the plane and his pride). He had landed with only 20 flap, and bit too fast. The only witness to this poor technique (who was not a pilot) asked me "aren't the brakes on the main wheels? They wern't on the ground, only the nose wheel". It would seem that the pilot was worrying so much about getting stopped, that he figured pushing forward on the controls would help. Nope!
I began to understand these concepts when, after what I though was a very precise landing in a C172, the instructor next to me said "nice landing, though a bit fast, do it again - more slowly this time" hauled the poor plane back into the air, and let go of the controls. With the power off, and suddenly 10 feet in the air, I got my act together fast! If I'd landed at the proper speed in the first place, it would have stayed on the runway, and just the stall horn would have wailed when he pulled!
Minimum takeoff airspeed with 10 flap is a soft field takeoff. It should be very well known to you as a feeling, not an indication on an instrument - 'cause you'll be watching outside the whole time! The plane will fly when it is ready. Even if the stall horn is peeping, don't worry, it's very safe as long as you maintain control. Once airborne, let it accellerate in ground effect. By the way, it's less than 50MPH.
Once you master the feel of the plane in the flare, and you are able to hold it there, and hold it off the ground, you will find landing other types of aircraft not too much a challenge. This technique works on most civil types.
Keep working to improve your skills, it will come more naturally soon,
Cheers, Pilot DAR