Speech is an underrated skill in Skyrim. It is one of those skills that doesn’t seem important until you get to a point in the game where there is a speech option that could completely change the outcome of the conversation. This skill can be increased by attempting speech challenges with anyone that offers one. Fortunately there is a glitch that allows you to get stuck in an infinite conversation loop where you can keep performing the same speech challenge over and over. This lets you quickly grind up the levels of your speech skill. Do this as soon as you can as it will be highly valuable to you throughout the game.
General bartering with stores does also increase your speech skill, but it seems to be a lot slower. There have been multiple occasions where I have purchased or sold something to a vendor and I have seen my speech skill increase. It doesn’t look like this can be spammed by constantly buying and selling the same items though.