This is basicaly a guide. Title says it all. Lynels are the most powerful monsters in hyrule and they take quite an effort to take down.
Recommended gear: One handed swords seem to work best, but if not two handers are pretty good as well. Don't use spears, as the whole point of a spear is to keep the enemy away and lynels don't recoil after hit. A good bow is recommended, especially triple shot bows. I recommend barbarian armour to deal maximum damage, and a lot of hearts, either from temporary or permanent.
Strategy: Hold out your shield and wait for them to make the first move, and DO NOT try and fight them from range. Their bows deal huge amounts of damage and they are deadly accurate. Get in close enough so when they attack that you can time a backflip, and if you time it right, you do flurry rush. after you've flurry rushed, take out your bow and quickly and immediately shoot them directly in the face. The face, not the mane – their headshot hitbox is quite small. if you headshot them, they'll kneel down for a few seconds on the ground. If ypu have a really good bow, keep firing, but before they get up just run up behind them and press a to mount. Then mash y to deal as much damage as possible. Rinse and repeat.
– If you're on a horse or have high enough ground, you can do the slow-motion arrow thing to deal tons of damage.
– To dodge the fireballs, just move sideways. You don't even have to sprint. After the fireballs you get a perfect chance to shoot them in the face for a mount.
– When they get down on all fours (sixes?) and run straight at you, sidehop instead of backflipping to activate flurry rush. Same goes for overhead strikes.
Finally, Here's a map for where to find these creatures. Red-maned is weakest, blue-maned is stronger, and silver-maned is strongest. What's not shown here is the silver-skinned lynel (with purple stripes), which is the strongest enemy in the game. I've managed to find it in the coliseum in central hyrule, but you may have to be so far into the game for it to spawn as a silver lynel.
Hope this helps you – happy lynel hunting!