Lane phase is a lot less dependent on you than it is on your support and opposing laners. The main goal is to stay alive, by any means necessary. Vayne needs gold, she can't get gold when she is dead. Get a feel for your support early, if you can trust them, take more aggressive trades with your tumble. If your opponent is a Draven or Lucian, don't trade. You won't win any of them if your opponent is good. Try to get your support to play an enchanter in this situation and keep the wave close to your tower whenever possible. Once you have ult, you can start to play more aggressively, but Draven/Lucian will still not be in 1v1 territory unless your jungler has gotten you a significant lead already. Tristana also outdamages you even after 6, but you can outplay her easier than Draven/Lucian by baiting her jump near a wall, tumbling opposite, and condemning her into the wall.

There are two ways I like to play the mid game with Vayne. My ideal way to play midgame is to stay bot and sidelane by myself, get some solo experience, and potentially get solo gold from T2 tower. This will only work if you feel confident in a 1v2 with either the opposing Toplaner/Jungler, or opposing Midlaner/Jungler. At this point in the game anything more than that is asking for trouble. If you do not have 2 items at this point, your best bet is to swap lanes with your midlaner. Farm midlane safely and get to your 2 item spike. Rotate for objectives and farm, farm, farm.

This is where Vayne shines the brightest. You should be looking for picks when possible, or forcing a 5v5 at Baron/Elder if your team/you have the lead. Ideally you want to play front to back in a teamfight. Kill the closest person to you first and go from there. If you can get in a position to kill your biggest threat first, go for that, but don't put yourself in a bad position just to kill the enemy assassin/mage. You have a team for a reason.