There are many ways you could approach this, but the best way is to find a qualified teacher. For something like this, even one lesson a month, (or even less frequently) would do much to help your tone along. Since embouchure and tone are such individual characteristics, both of you and your equipment, I can only offer a few sweeping generalizations.
In my experience, especially in younger/newer players, there is a misunderstanding when it comes to playing exercises, including long tones. Simply playing long tones will not make your tone much better. What you want to do is play long tones while isolating the good portions of tone. Listen carefully to your tone, and morph it so it sounds the best it can be. You won't hear an immediate difference overnight, and the changes will be subtle, but the key is to LISTEN. This rule applies to any exercise you do. I also recommend slow lip-slurs (note changes without using your tongue) as a good strengthening exercise. Again, do them slowly, while listening, and don't accept a bad sound or execution. Your goal is to learn the cleanest, prettiest ways of playing a particular passage, and move the percentage so it sounds like that every time you play it. This takes time.
Breath is important as well. I won't go in depth as there are countless good sources of breathing information out there. It's essentially the same for any musician that uses air, but I highly recommend checking out breathing tips for singers. Singers tend to concentrate more on a relaxed, tension-free breath than many instrumentalists do, and I advocate that technique. Most importantly, low and slow, filling up your abdomen. Your shoulders and chest should not move much, if at all; however you should feel and see your stomach expand.
Embouchure questions really should be directed to a competent teacher who can watch and hear you play, but here are a few things you can experiment with before heading to one:
Try putting more of the top-lip in the mouthpiece, and less of the bottom lip. Try vice-versa. Make sure the mouthpiece is relatively centered on your lips from side-to-side. (Assuming you don't have problematic teeth, as I did in the past). Try keeping your same embouchure placement, but concentrate on focusing your airstream down, up, or straight down the tube. (Alternate). Try puckering more, or less. Try pointing your instrument more down, or more up, in gradual increments, without moving your head to adjust the pressure from the top to bottom lip. Make written notes of what works, and preferably record or videotape your playing.
Lastly, as much as I hate to advocate equipment changes early, if you are playing on a 12c mouthpiece, try obtaining a 61/2AL size mouthpiece. Everyone's lips/mouth/face/playing habits are different, and the idea of a one-size-fits-all mouthpiece is ludicrous. This should be tried after the above steps.
Again, my advice is to find a teacher. However if you have a good ear, and you spend some time at it, simply experimenting with some of the things above may help you.
PS. I didn't address the high-note straining you mentioned, as it's not necessarily related to your tone issue. However your lips should not be doing most of the work. Above all don't use mouthpiece pressure on your lips to get the high notes! Try whistling a low note, skipping up to a high note, then back down. Notice your tongue goes from "oo" or "ah" to "ee" this extends to playing too! Focus on that change, then concentrate on blowing faster air. Less air, but at a higher speed. You'll get it, but it takes time. My advice is concentrate on the tone issue first.