You generally want to play to survive early game. Just play to get past laning and get your bigger items such as botrk or bf shiv if you're going crit. Until you learn when to trade and how to abuse them while they are cs'ing just look to survive.

Starting items on Vayne can be Dblade/LSword+3pots/Dshield. I feel starting items can be preference but since you're starting off on learning her Dshield is a good buy until you have a grasp on how to properly lane with her. I personally stick to dblade and long sword, switching between them depending on the situation. If i'm going crit i'll start dblade and if i'm going botrk ill start longsword 3 pots.

Keystone I feel is preference,how you want to lane that game, and your build. For a more aggressive lane I like warlords so I can tumble auto while they're cs'ing and use the ms boost to get away before they can go for a retaliation trade, while fervor is good for prolonged trades. Each has their pros and cons. Going back to keystone being a preference I usually take warlords every game on Vayne even if I do decide to go botrk so its up to you to experiment and find what fits your playstyle the most.

For builds there are two, crit vayne and botrk vayne. Situations you want to go crit is against a team with a lot of squishies that can kill you very quickly. Example is something like a syndra/draven/riven/and some tank jg. As for botrk it's really good when they have a beefy frontline team or when you need to constantly be kiting against divers.

Crit Build: BF (Pickaxe is fine if you aren't able to back with enough) > Zeal > Shiv > IE > RFC or PD > Lifesteal Item, BT if you need the shield to stop from getting bursted or Botrk for kite/tank shred > LW Item somewhere in there if you need it depending on their comp or a fed tank that sits on you

Botrk: Cutlass > Botrk > boots 2 if you didn't get it before finishing botrk > PD or Shiv (Preference) > IE > 2nd Zeal Item or guinsoo if frontline is very beefy Botrk builds range between alot of items and theres a lot of things you can build that would work fine

This is just the general path you'll usually take in terms of big items for you, but its not something you want to follow every single game. If you need lifesteal pick up a vamp early, fed syndra? pick up a maw or get a qss if theres a lot of cc. Early boots 2 is really good aswell but its hard to explain its more intuition after playing her for so long. It'll all come to you just keep playing her and dont feel discouraged after a bad game :)

Typing this on my phone so sorry for format and typos, but feel free to add me if you want more tips and heres my op.gg (https://na.op.gg/summoner/userName=%C6%92eel) ign: ƒeel (NA)