Since someone requested a Wyvern walkthrough, I thought I'd put one together. This won't be a terribly elaborate guide and it certainly won't cover all of the various strategies that can be found over on the Wyvern page. It will, however, hopefully lay out some good guidelines to follow when fighting the Wyvern for the first time.
For fighting the Wyvern, I'm going to be using the Crystal Storm spell. It's one of the top-tier weapons in the game and is fairly easy to obtain after defeating the Wall of Flesh, although collecting Souls of Light can be a bit annoying. There are many items which can be used to power up Crystal Storm, but all I'm really going to be taking advantage of in this walkthrough is the Deadly prefix on Crystal Storm, the Clairvoyance buff from the Crystal Ball and the Sorcerer Emblem. Other items such as the Magic Power Potion or Bowl of Soup can make spells even more powerful, but they aren't really needed against the Wyvern.
Another great weapon choice is the Star Cannon, provided one has enough Fallen Stars to use for ammo. Cursed Flames, Repeaters with Jester's Arrows, Demon Scythe and Dao of Pow are also excellent weapon choices against the Wyvern, especially because they take advantage of the fact that the Wyvern has multiple segments and these weapons can damage many segments simultaneously if aimed well. Damaging multiple segments with each shot can cause the Wyvern to die very, very quickly. Some other weapons such as the Magical Harp, Flamethrower or Megashark are obviously good but require beating some Hard Mode bosses before they can be obtained.
There is one last thing I should mention here. The Mana Flower works extremely well when used with Crystal Storm and I absolutely recommend using it. I will not be using the Mana Flower in this walkthrough, however, so I will unfortunately have to drink mana potions manually. By the way, when using spells I always carry around a stack or two of Greater Mana Potions which can be purchased from the Wizard.
When fighting the Wyvern, one should aim to have a minimum of approximately 45 defense. The way I'm going to accomplish this is by wearing the full Cobalt Armor set with the melee helm, by Reforging my accessories with some defense bonuses, and by using a couple Ironskin Potions. Of course, if you want to one up me by using Mythril Armor or even Adamantite Armor for your first encounter with the Wyvern, that's a perfectly viable option as well.
Not that much healing is required for fighting the Wyvern. Honestly, the battle should be over in just a minute or two if you're doing it right, so you won't really need more than a handful of healing potions. I, however, have a habit of always carrying around a stack of Greater Healing Potions =P
As far as accessories go, a minimum of Spectre Boots (flying + running) and Cobalt Shield (no knockback) are required. The Shiny Red Balloon, Cloud in a Bottle or Cloud in a Balloon are also extremely useful here, but I won't be using them. A Lucky Horseshoe can be convenient for eliminating fall damage should you accidentally decide to go skydiving, but that shouldn't be happening. Some other good items to have in your inventory are a pickaxe, some building materials, some sort of grapple hook, and possibly a Depth Meter or GPS for calculating altitudes.
Slaying The Wyvern
You can fight this guy pretty much anywhere, but I personally prefer just building upwards until I get to an area where Harpies spawn at. If Harpies are spawning repeatedly at a certain altitude, then generally speaking the Wyvern can also spawn there as well. What I want to do is just build a little flat area that I can use as my "arena" so to speak. You can take this to the extreme and build a whole skybridge if you so desire, but it's not really necessary.
Wood Platforms are a great choice for building material, as you can shoot through them and you can drop down through them as well. As you can see in the picture above, my health is rather low because Pixies and Harpies and such have been beating the crap out of me while I'm building my bridge or whatever. You know what I'm going to do before I start fighting the Wyvern? I'm going to log out and back in, heal with the Nurse, and then return to my arena and start the battle with full health. That same thing applies if the Wyvern spawns before you feel that you're done building your arena. If you don't think you're ready to fight the Wyvern just yet, you can always log out of the game (Wyvern will despawn) and then log back in and continue whatever you were doing.
You've been wondering about those red arrows I drew on the screenshot above, haven't you? Those represent the Wyvern's attack angles. If you're standing still, he will always approach from the upper left, and then the right, and then the left again. The Wyvern's movement patterns form sort of a distorted figure eight. This fact combined with his speed means that the key to dodging the Wyvern is vertical movement.
Well, after logging back in again and waiting for some time, the Wyvern has finally spawned! I suppose I should get to the point where I explain my strategy for fighting him. First, I'm going to be spamming Crystal Storm in his general direction the entire time. Generally you want to aim slightly ahead of his path so that most of the shots hit. When he's off the screen and you can't see him, fire in the direction of those red arrows that I drew.
Now, when the Wyvern comes charging in, I'm going to simply jump over him. This is really easy to do once you've practiced it enough, but it might take a while to learn the timing. Ideally, you want to jump at the last second possible, which is approximately when the Wyvern is about as far away from you as he is from me in the picture above. Since the Wyvern moves very fast, playing the game on fullscreen in the highest resolution available is extremely helpful for spotting the Wyvern's incoming attacks. The reaction speed required to jump over the Wyvern is moderately high, but anyone can pull it off with a little bit of practice. Once you've jumped and dodged his head, you want to use the Rocket Boots to fly/hover for a second while the Wyvern passes by.
If you need to drink a mana potion while using the Rocket Boots, feel free to let go of the jump button (Spacebar by default) in order to press the Drink Mana Potion hotkey (m by default). The Wyvern's body doesn't do nearly as much damage as it's head. While it's ideal to completely dodge the Wyvern, it's acceptable to get hit by the body if you absolutely need to drink a mana potion so that you can keep spamming Crystal Storm. Of course, if you have the Mana Flower then this isn't an issue.
Suppose you know that the Wyvern is going to attack from the right next (since he previously came from the left). One little trick that I advise doing is running to the left (away from the Wyvern) a little bit before you think his next attack will come. This gives you extra time to react for your jump. If he's coming from the left next, you would instead run to the right a little bit.
Another option available to you for dodging the Wyvern is dropping down through some wood platforms at the last second, instead of jumping. You can then use a grapple hook or Rocket Boots to get back up.
With your reasonably high defense rating (45+), you can still afford to take a few hits, so don't worry if your dodging isn't perfect. Just do the best you can and most importantly keep shooting at the Wyvern constantly, even when he's off the screen! The battle should be over in about a minute or two, and you should only have to dodge the Wyvern about a dozen times or so before it dies. If your encounters with the Wyvern last longer than that, then you know that you're doing something wrong. The Souls of Flight that it drops will appear wherever the Wyvern's head was when it died. If that location happens to be offscreen somewhere, you might have to build a little tower/bridge or fly around a bit in order to collect the souls.