Welcome...

Veigar is unbelievably op...like he can just one shot you late game and most mages usually can't do that but enough of rambling lets just get into it

Flash: No going wrong with flash especially to get people caught in your E

Ignite: Kinda just a safe bet if they're low early game Flash + ignite and back for quick so you can be ahead of them

Any player will kinda realize what's going on here, I'm building AP why you ask Veigar's ability literally gives him more AP!

Well lets just forget that sentence you just said and say, the more the merrier!

The order of the items are there because of the rune "Cut Down" this will mean that you can pretty much kill...any champion, it's because of the HP you could even go top and contest with champions like Garen cause it's a simple W-Q-E and run and they're maybe 1/4-3/4 of their HP
1. I'm not saying you HAVE to use cut down but, simply put it you can (Last Stance And
Coupdegra easily could be used)
2. These are just the runes I use, mess with them however you want

Speaking of runes...

The Sorcery runes all have their reasons

Arcane Comet: Building AP will make it do more Damage anyways

Manaflow Band: You will use A LOT of mana early game

Transcendence: More AP :p

Gathering Storm:... MORE AP

I know this sounds stupid but bare with me, finish off minions with your Q, NOW NOW...With most champions you don't want to use mana while in lane which is true but veigar gains 1 AP For a small minion and 2 for big ones so basically just use it when there are low minions

UNLESS there's a champion who counters Veigar very well you would want to keep a portion of you mana enough to use your E so you can run away from them

REMEMBER THIS IS USUALLY WHEN VEIGAR HAS A LOT OF AP SO EARLY GAME THIS MIGHT BE A BIT USELESS

The attack pattern for veigar is one of three which I like to call

All In: E-Q-W-R
Damaging: E-Q/W-Q/W
Skill:(This will be only for riskier plays and usually consists of pushing a bit)
Q-E-W-Q-(R)
Now you might ask why it's risky simply put unless the champion is SUPER pushed they usually will stay a bit back in lane so you might have to push them in order to stun them and get the full elimination because they usually back off after the Q (Basically asking for a gank to happen)

Veigar is quite good at either

1. Saving people with his E

2. Picking off high health pre occupied full hp champs (With high AP)

THAT'S IT...now go play with him..if you want....