Elise is one of the best early game duelists in the game and can easily duel other strong early champs such as Lee Sin or Kha'Zix. She has strong ganks and thanks to her Rappel, she is a potent tower-diver.

Skirmisher's is used to further increase Elise's dueling power, and Runic Echoes + Rylai's gives alot of damage as well as a slow. Elise's spiderlings apply the slow, which makes it easy to keep people slowed. Dead Man's and Abyssal both offer good damage with tankiness along it. Decide on which one to go earlier depending on enemy comp (ad or ap heavy, who's fed)

vs aa heavy ad team go Tabi.
vs cc heavy magic dmg team go Mercs.
vs lots of slows go Swiftness.
If the enemy team has a balanced team, choose on your boots depending on who's causing most trouble to your team (etc their Vayne is fed and others are weak=>Tabi)

Sorcs are ok if you're super fed, but otherwise I wouldnt recommend them as much as the ones above

First think of whether you want to itemize vs ad or ap. If ad, do they have a lot of crit? If yes, go Randuin, otherwise Frozen Heart. I usually prefer FH, unless the enemy comp has something like GP top, Yasuo mid and MF adc. If you want to itemize vs ap, go Locket or Banshee. If your team has no Locket and support won't build it for some reason, go Locket. Otherwise Banshee.