Whenever I play multiplayer, I start at the beginning of the module, by myself, away from everyone else. Is this supposed to happen? Because I'm wondering if that's what happens on the Undermountain mod.

<font color=skyblue>There is always a starting point where everybody will begin the adventure. I do not know of a work-around for that. The Undermountain Project is the same. Everybody has to start in the tavern, and grab a pass to be able to enter the door that leads to Undermountain's caverns. In order to meet up with people, you gotta run to catch up, make them wait on you, or have a planned time to meet at the start point.</font>

If however you enter a server, leave and then re-enter, the server will remember where you last where and place u back there *providing* you are entering under the same character. If not the server will reset you to the module starting point.

You almost always start at that starting point, and it's unlikely there will be many others around, yes. There are a few exceptions; some modules will let your starting point change after the first session, or after you've reached a certain level, or when you play a certain class, joined a certain guild or temple, and so on, either teleporting you there straight away or right after you log in. But usually that's a preset starting point too.
It used to be the case that you would only jump to the place where you logged out if that logging out happened with the same character in the same server session (for example, if you've logged out and right back in, or if the server is a persistant world which doesn't know any downtime nor needs resetting. If the server crashes or is reset you begin in that starting area once again).
I believe that with Bioware allowing builders to access certain files it is now possible to store a player's location (and state) and carry the information over from one session to another, but there are only a handful of mods where it is successfully applied.

Usually it shouldn't be too hard to get from the starting area to the place where all the action is, and it'll give you a chance to reread the server rules and background.