Out of all the bosses Pre-Hardmode, skeletron gave me the most difficulty. His hands do a ton of damage, and later on his projectiles can be very hard to dodge.
Eventually I figured out that his attacks are pretty easy to learn, and you can avoid them with the right strategy.
The short version is that you use a Molten Fury and a Space Gun to beat his various phases. There are a few things that need to be done before you start the fight, so I'll explain everything you need to know.
Preparation
So what you’re going to need for this fight is Molten Armor, a Molten Fury, Space Armor, a Space Gun, a Shield of Cthulu, Rocket Boots, and the best movement accessories you can get. I used Terraspark Boots and a Horseshoe Balloon, which helped me dodge attacks and avoid fall damage. I also recommend using a potion like an Iron Skin potion.
Next, you’re gonna want to build a decently long bridge right next to the dungeon. It doesn’t have to be too long, but should be a good enough area to run and avoid his attacks. It’s also important to put a wall at the end of the bridge to keep you from falling off. Then, you should place campfires and Heart Lanterns with enough range to cover the entire bridge.
You can check the range by looking at the buffs under your inventory and seeing when the buff disappears.
For arrows you should use Jester’s arrows since they can hit both his hands at once.
The Fight
You should start the fight with the the Molten Armor in your armor slots, and the meteor armor in your social slots.
The fight should be started as early in the night as possible, or else he’ll turn into a dungeon guardian and one shot you. When you’re ready to summon him, talk to the Old Man at the dungeon. The first part of the Fight is definitely the most difficult because of his arms.
You want to keep firing the Molten Fury towards him while running away from him and dodging the hands. Don’t bother shooting at his head yet, but work on damaging both hands at the same rate.
When one hand dies, he immediately starts firing skulls at you, so you want to break them both hands around the same time.
As soon as both of his hands are dead, open your inventory and switch to meteor armor. You can quickly do this by right clicking on it when it’s in your social slot. For the rest of the fight, you want to go in circles around his head while shooting it with the Space Gun.
The easiest way to dodge the skull projectiles is to fly straight up and then go over his head. You just need to keep doing this until he dies but it won't take very long to beat him like this.