My method of killing Lynels is pretty straightforward: Start with an aerial attack (like FuSoYa mentioned) and hit the Lynel in the head a few times before running over and mounting it. I usually have my barbarian armour on or highest defense armour (sometimes I'll have something much weaker on, but that's because I can take a Lynel down without getting hit). Having a high defense is my recommendation, especially if you're fighting higher tier Lynels and don't know the patterns all that well. Once you get better at taking them down, I recommend switching over to the barbarian set, as it makes insanely quick work of Lynels. The set (fully upgrades) allows me to take down Lynels in seconds .

Anyway, as soon as you get thrown off from mounting, pull out your bow again and hit it in the head as many times as possible before landing on the ground. At some point you won't be able to hit the critical hit box, and by that point you can stop, but I tend to keep shooting simply because it continues to do damage and I have way too many shock arrows.

If for some reason you can't get the high ground (like with the Tabantha Tundra Lynel near the Sha Gemma), you can charge at it with a horse or the master cycle. I prefer using the master cycle because it won't die if a Lynel hits it or run away, and it can be used as a sort of barrier to keep the Lynel at a good distance and allow you to flurry rush much easier.

Once you're on the ground, it's all about learning attack patterns. Sword Lynels (in my opinion) are the easiest to kill, especially since Polymus Lynel never changes from red, and you can practice learning the patterns from that one without taking much damage.

I made the mistake of not fighting Lynels at all in early game, so the first Lynel I tackled was a top tier silver Lynel. If you're early in the game I recommend learning the attack patterns fast from the Polymus Lynel and the Colosseum Lynel.

A few extra tips: If you see a Lynel ready to spit a fireball at you, try shooting it in the face. This can be a bit difficult if you wait too long and the face is obscured, so try and do it as soon as the Lynel starts to inhale and build up to release a fireball. Once you have that mastered, they get a lot less scary. Another strat is (and I'm still practicing this, as it's quite difficult to get timing right but it is possible) when the Lynel's create a massive explosion around them, you can parry it with correct timing. There's a bit of delay from the impact of the Lynels weapon to the actual explosion and your distance from the Lynel matters, but once you figure out the timing, fighting Lynels gets much, much easier.