Doom 'n Gloom
Doom 'n Gloom is Vex's passive, and it has two parts to it.
DOOM: Periodically, Vex gains Doom, which causes her next basic ability to interrupt dashes and fear affected enemies, while also being slowed.
GLOOM: Nearby enemy champions that dash or blink, as well as targets hit by Looming Darkness, will be marked with Gloom for 6 seconds. Vex's next basic attack, Mistral Bolt, or Personal Space against a marked target will consume the mark for bonus damage, and refund 25% of Doom's cooldown.
This passive is pretty amazing and will definitely be the reason Vex sees play. Most of the meta champion pool on mid lane (and in general, to be fair) all have at least one form of dash, and this will allow you to not only interrupt those dashes, but gain bonus damage, allowing you to surprise the enemy with more burst damage. You will also notice when your Doom passive is available by the bar underneath your mana bar, and your abilities take on a light green colour.
Basic abilities (not your ultimate Shadow Surge) will apply Doom, so you can use Shadow Surge to set you up and get yourself in a good position without wasting the fear, then use a follow up ability to land the fear. I recommend Personal Space as it has an instant cast and is not a skill-shot, but rather an area around you, and this will allow you to follow up and hit your skill-shots Mistral Bolt and Looming Darkness easily.
Mistral Bolt
Mistral Bolt is Vex's bread and butter ability. Vex unleashes a wave of mist in the target direction, dealing magic damage to enemies it passes through and accelerating in speed, though reducing in width after travelling 500 units.
This spell is important for Vex - it's her wave-clear and main damage source in one spell. Learning the range of this ability is really important because it's quite slow initially, and only speeds up once it passes 500 units. You can also use this ability to proc Gloom from a long range, so one of your harassing combos is opening with Looming Darkness into Mistral Bolt.
This ability has a low mana cost early, so make sure to harass your lane opponent with this constantly!
Personal Space
Personal Space is Vex's second damage spell. On activation, Vex emits a shockwave around her, gaining a shield for 2.5 seconds and dealing magic damage to all nearby enemies.
Personal Space has some interesting AP scalings - the damage only scales with 30% AP, but the shield strength has a HUGE 80% AP scaling! This is one of the biggest shields in the game, especially for a carry champion, and will allow you to soak a lot of damage if timed correctly - however, the mana cost is 75 from level 1 onwards, meaning you have to use this ability sparingly to not run out of mana, and it also has a 20 second cooldown early game. Oof.
The main use of this ability after laning phase will most likely be using it to proc Vex's Doom passive, you can use your ultimate Shadow Surge to engage with, and because her Doom passive only works on her basic abilities, you can land a very easy follow up fear using Personal Space - as it's the only ability in her kit that is not a skill-shot, allowing you to hit the rest of your abilities with ease.
Because this ability doesn't do much damage, but its shield scaling is huge, we'll more than likely max this ability last.
Looming Darkness
Looming Darkness is Vex's next ability. Vex commands Shadow to fly to the target direction, increasing the radius of the area as it travels. Upon arrival, Shadow deals magic damage to all nearby enemies and slows them for 2 seconds.
Looming Darkness also applies Gloom to effected enemies hit. Because of this, a great harassing lane combo is to open with Looming Darkness into Mistral Bolt, because Mistral Bolt also consumes Gloom, which makes for some great harass. Also, you don't want to be using this ability last in your combo simply because of the bonus damage from Gloom, unless you can use an auto attack to proc it.
You want to max this ability second, as this abilities AP ratio increases with level ups, and it's also your second most consistent damaging ability!
Shadow Surge
Shadow Surge is Vex's ultimate. Vex commands Shadow to launch itself in the target directio, granting sight of the area along its path for 0.4 seconds each and dealing magic damage to all enemies hit.Shadow stops upon colliding with an enemy champion, marking and revealing them for 4 seconds. Shadow Surge can be recast while the target is marked.
The recast pulls Vex towards the marked target while being unstoppable, and upon arriving, consuming the mark and dealing heavy magic damage. If the marked target dies within 6 seconds of being hit by Shadow Surge, Vex can recast this ability again at no additional cost for the next 12 seconds.
This ultimate allows you to do crazy things with Vex! While the ability seems simple on paper, it's complex in practice - as its a thin, slowish skill-shot that grants you resets, it's very important to not waste this during a fight. Prepping your enemies with Doom 'n Gloom's fear passive gives you can easy route to landing this ability, however you have to be sure that they will die within the next 6 seconds. Also, you don't want to randomly throw this out and not follow up, as the huge potion of its damage is on the impact damage when you dash, not the initial damage, and the reset is insanely important to Vex's team fight potential. This ability could win or lose you a fight based on the amount of resets, so be careful!
After testing Vex a lot, Personal Space is BY FAR the better max 2nd. It does a lot more base damage, more shielding, and lowers in cooldown when leveling it up, and Looming Darkness only barely increases in base damage, increases AP ratio which matters more late game, and doesn't decrease in cooldown. When you're resetting with Shadow Surge, what's the most important follow up spell? Personal Space. Gives you a huge shield to help you last in fights to get another reset, and it gives you an instant fear from Doom 'n Gloom. You'll always be able to cast another Looming Darkness, but you need Personal Space to be available with as many dashes as possible.
Mistral Bolt > Personal Space > Looming Darkness, with a level in your ultimate Shadow Surge whenever you can at 6, 11 and 16.
At level 1, you'll almost always start with Mistral Bolt. It's her bread and butter - damage, range and waveclear. She has no CC in her basic abilities, but in her passive Doom 'n Gloom, so she can get a 5 man fear with Mistral Bolt should you choose to invade.
At level 2, I highly recommend Looming Darkness, as its damage will be a lot easier to proc than Personal Space's, and it also applies Gloom for bonus damage. Looming Darkness followed by Mistral Bolt makes for huge damage level 2. You can also take Personal Space if you're vs. a champ that dashes as early as level 2, as it's an AoE instant interrupt with your passive.
At level 3, we can put another point into Mistral Bolt if we think we aren't being ganked, or the enemy mid laner can't get in range of us or land lots of poke, or we can level up Personal Space to have all 3 of our abilities at level 3, be a bit safer, or if you're vs. a hyper mobile champion like Yasuo, another way to proc Gloom.