Q>E>Q>W>Q>R..... max Q first, W second (but grab E at level 2)
Only last hit minions and farm safely untill level 3, then start harrassing with your Q and try not to waste mana, W shouldn't be used at all during laning phase except to maybe combo the enemy. Harrass untill the enemy has around 50-60%hp, if they do and they don't go back they're dead, as soon as you land your stun unleash your fury, E>Q>W>R>W>Q if they flash Rappel after them, use ignite to secure kill and if you're chasing as spider and they out run you or disengage then R again and Q them from a distance. BAM dead, this can be done as soon as level 4.
Go b and upgrade your boots and start building Haunting Guise, grab more pots and wards, mana pots are really important in Elise because you aren't building any mana items and you took the defensive masteries. Back in lane repeat process. --- Signature for rent
21/9 to be offensive but 9/21 lets you be a bit tankier (what I prefer with so many bruisers top). --- You like it or you don't, but i hate people saying it's 'an acquired taste." That's just a cop out. - bookwormbabe29 on beer.
Red: Hybrid pen or MPen Yellow: flat Armor Blue: flat MR (unless the enemy top, mid, and jungler are all physical damage) Quints: flat AP, hybrid Pen, or MPen
I would use hybrid Pen against a melee that you can get more AAs on, or MP against a general opponent. I'm not sure if flat AP quints are worth it on Elise, probably not unless you anticipate having a hard early lane, and you want the damage on the spiderbomb.
I use 9/21/0, but 9/0/21 sounds vaguely interesting. There's a lot of wasteful masteries in 21 utility for Elise, but there are some moderately interesting ones (GP10, summoner cooldowns, MP5 -- the rest don't seem useful for Elise). The utility tree could be interesting for an Elise with a teleport for map control. Teleport CD + Distortion boots, etc. Good times.
In terms of starting build, I would get a flask on Elise because flask. Guise/Sorc boots immediately followed by whatever. "Whatever" for me would usually include a SFC and Liandry, finishing up with Rylai and Bulwark (earlier Bulwark if necessary / if enemies have good magic damage, earlier Rylai if SFC is irrelevant, early Doran Shield if necessary). Abyssal also a theoretical pick after Rylai if the enemy is heavy magic.
In terms of item theory, Guise / SFC / Bulwark are all better early, and Liandry / Rylai is better late.