It sounds like you need a quick overview of some basic ASP.Net Webforms concepts. Up first I'll counter a common newbie misconception:
Your Page class does not hang around on the web server for very long
I think many new ASP.Net developers have this idea of the web server keeping a single instance of their page class for every user session that hits their site, and each postback or event uses this same page class instance. That's just not how it works. ASP.Net page class instances are nearly always created and destroyed again in well under a second, and most experienced developers see it as a big problem if it takes longer.
ASP.NET relies on the HTTP protocol
The thing to remember here is ASP.Net still relies on the HTTP protocol, and http boils down to requests and responses. When you view a web page, your browser first sends a request to a server. The server responds, usually with an html document. The browser will then parse the html; based on what it sees in the html the browser may send more requests to the server for additional resources, such as javascript, images, or css files. Each request results in a separate response, and the browser uses all these resources to render the page to the screen. However, the ASP.Net runtime normally does not have to process the additional requests (that would make things slower) — ony the initial html needs ASP.Net support; you want the other resources to be basic files that can be cached.
The ASP.Net runtime creates a new instance of your class for every request.
When the ASP.net runtime processes a request for a page, it will create a new instance of your page class. The runtime will follow the ASP.Net Page lifecycle (this should really be named the "ASP.Net Page Request Lifecycle"), and call certain methods or raise certain events in this class instance, in a specific order defined by the lifecycle.
This means every postback or event runs in a different instance of your class.
It also means every postback or event is rebuilding and transmitting all of the html the goes into your page, and not just the portions you want to change. For your server code, the consequence is the only thing class-level variables are really good for in ASP.Net is things that will be used within a single http request. For the browser, the consequence is you're working with a brand new DOM after every event.
To understand all of that, it's important here to also have a good understanding of the difference between a class and an instance of a class. A couple items in your question make me unsure whether you have this understanding yet.
The ASP.Net runtime shares one application instance among all users of your site
The web server typically only has one instance of your application for the entire web site and all it's users. Therefore, anything with a Shared/static scope is common to every user. It's rarely appropriate in ASP.Net for anything to be Shared/static.
So how do you handle data that should live with a single user or visit to your site?
This is exactly what the Session is for. A session will always be unique to an individual request at any given time. You're worried about multiple functions accessing the session at the same time, but this does not happen. The ASP.Net Page Lifecycle ensures that unless you manually spawn additional threads, only one function at a time is running for a given HttpContext and Session. If a user somehow sends two requests at about the same time that should have the same Session/HttpContext, one will be held by the ASP.Net runtime until the other is completed. If you don't want to reference the session all the time, you can build properties in your class that wrap session variables. See @Pankaj's answer for an example.