Dodging skillshots can be one of the most satisfying feats to pull off in Heroes of the Storm especially when you did it consciously. It sets apart a good Jaina from a great Jaina simply by saving herself from taking what can sometimes seem to be negligible damage. The more important reason to dodge skillshots, aside from avoiding damage, is because many skills shots can stun or root your hero. A proper stun or root can get you out of position and allows the enemy team to quickly engage your hero. Channeled abilities that get canceled by stuns like LiLi's Jug of 1,000 Cups is another reason why you want to dodge skillshots.
There are three ways to improving your ability to dodge skillshots: watching hero animations, anticipation, and mouse positioning. Before I go through these three topics I think it goes without saying that keeping your hero moving is one of the best tools for not getting hit by skillshots. Even if you are auto attacking as Raynor, take the time to take a step from side to side in between each auto attack, just to make it that much harder for the enemy team to hit their skill shots on you.
1. Watching Hero Animations
Each hero has a specific animation for their skillshot casting as well as their respective skillshot. Watching for these animations can greatly aid in dodging a skillshot coming your way. For example, Muradin raises his right arm that is holding a hammer when casting the skillshot and this is followed by the skillshot itself which is animated as a lighting hammer traveling forward. The skillshot casting animation time and the skillshot itself can either be fast or slow. Anub'arak's Impale ability has a slow casting animation time and a slow skillshot animation time, making it easy to dodge an Impale coming your way. Zeratul's Singularity Spike ability has a fast casting animation time and a fast skillshot animation time, making it difficult to dodge. Some casting animations are also easier to spot than others. Arthas, for example, has a very distinct animation for his Howling Blast ability.
It takes a while to get used to watching for hero animations and I suggest playing a few heroes you rarely play just to get used to their animations. I also recommend picking one enemy hero during the loading screen that you will consciously watch for when you are ever in a lane with them or even during team fights.
Zeratul's Singularity Spike: Fast Casting Animation & Fast Skillshot Animation
Anub'arak's Impale: Slow Casting Animation & Slow Skillshot Animation
Many heroes have skillshots, and simply reminding yourself before the start of the game that the enemy will try to hit you with them will immediately improve your consciousness to dodging skillshots. Sometimes it is so much easier to just run when you are getting chased down by the enemy, but always ask yourself if anticipating to dodge a skillshot is worth it. I'm not talking about zig zagging all the way to safety and just giving your enemy more time to catch up, I'm referring to actually positioning your mouse so that you are ready to dodge (more on this later). Watching the hitbox direction (the little pointer underneath each hero) can also help you anticipate if an enemy hero is intending to shoot your face with a skillshot. Finally, although you are not technically “dodging a skillshot”, hiding behind minions and structures is also a great way to deny your enemy from hitting you.
Although not technically a skillshot, the ability to dodge Kerrigan's combo is a useful skill to have
3. Mouse positioning
So you knew that Kael'thas was going to stun you with Gravity Lapse, but you couldn't quite react fast enough and he stuns you and blows you up with a flamestrike and two living bombs. What can you do to not let this happen again? A lot of the time it comes down to mouse positioning. If you are not casting abilities or throwing down your own skillshots, your mouse should be positioned above or below your hero, never in front or behind. This allows you to move your hero laterally (side step) much quicker and react to an incoming skillshot much faster. Here are a two images to illustrate the ideal mouse position:
Mouse Positioning for Slow Reaction Time
Mouse Positioning for Fast Reaction Time
Animations, Anticipation and Mouse Positioning, that's all there is to it. Dodging skillshots definitely requires some practice and I even suggest going on Try Mode to see the different animations. Soon enough you'll be side stepping like a pro and maybe even learn how to dodge giant balls of fire.