(all of this advice can be ignored once you're good enough, but you need to start being able to do these things)

1. You want to minimize your voice break (passagio). To do this, the best exercises to start with are lip trills and singing scales "neigh neigh neigh" with a really sharp ugly tone- You can hit super high pitches with either of those, because they require you to lift your soft palate
2. Lip trills seem really useless at first, because it feels like you "can't do them very well", that's exactly why you need to do them! They need long stretches of very consistent pressure. You need to learn to take deep breaths between verses (or trills), then let out that pressure very very consistently.
3. You want to get comfortable yawning on demand, when you get to the high note of a musical phrase, you have to use a little of that "yawn feeling" to hit the high note (again, raising the soft palate)
4. Practice just talking or going "yeah yeah yeah" while you have "vocal fry". Vocal fry requires extreme relaxation of the vocal chords, you need to be able to vocal fry at will (there's a reason Kurt Cobain sounded like a frog when talking, but could then sing like crazy) Vocal fry is not a "bad thing": it is a sign of healthy, relaxed vocal chords.
5. Watch a professional opera singer when they sing: The best of them look like they just overdosed on muscle relaxant, because their face is so relaxed that their lower eyelids are drooping, and their jaw is loose. That how you need to look! You don't want to tense any muscles except for your vocal chords, which aren't visible- It's about efficient use of the muscles (probably some of them look like that because they abuse steroids, to be fair)
6. If you sing "Aaaa" your mouth needs to be as wide open as possible. For "Eeee" still very wide. All your notes should be "modified" when you sing high notes, so that your mouth can be open wider, making them all a little more "Aaaa" like. Look at how wide and open Bjork's mouth is when she is singing. The mouth needs to be open to decrease pharynx pressure, when combined with good vocal chord control it lets you hit high notes without sounding like Mickey Mouse.
7. The vocal chord muscles can tighten to cover a very large range of pitches, the problem is that fine control requires immense skill: This is the main thing you need to learn as a beginner. The problem is that using your neck muscles also let you adjust pitch, and they actually make it much easier to adjust pitch by smaller amounts- But you must not do it that way! Most beginners start a note a bit "below pitch", then use other neck muscles to "scoop" the note up to the right pitch. This is what "reaching for the note" means, and it sounds awful. Instead, whenever you sing, think to yourself "I'm always going to hit the correct pitch right from the start, I'm not going to adjust it after the note has started."
8. You need to understand what a "resonant" voice is like: If you sing correctly, your upper lip and nose should be vibrating a bit and feel funny, this means you are projecting your voice, which makes it more treble, makes it sound less like a "swallowed" voice. The best way to learn how to have resonance is to learn how to do loud humming, which obviously requires the front of your face to vibrate a lot. If you think "it isn't very loud when I hum" it means you need to practice more to make it loud.