Being a new web developer can be hard. There are way too many technologies and frameworks to learn and it is very easy to get lost in the sea of buzz words and new things. Below is a quick list of core areas an Asp.Net web developer should look into:
1 - WebForms - Although you might want to go down MVC route - no pun intended, which is currently an alternative to WebForms, being an Asp.Net developer I'd dare say you will come across webforms inevitably. WebForms is a beast! Server controls, user controls, page-life cycle, state management of controls and so on. ... and don't forget Asp.Net Ajax. Now that's another beast on top of webforms.. What I am trying to say is, they are great component frameworks abstracting underlying lower level web technologies (HTML, HTTP etc..) but they have a steep learning curve after being able to code 'hello world'.. and they are not the Web! So don't despair! It will take time.
2 - Javascript - It is a proper language. Not feature complete, or even 'really' object oriented, but it is a programming language. So take it seriously. Today with AJAX and Rich interfaces users expectations are high and we have to meet them. The only way is, a good understanding of javascript and DOM. I recommend taking sometime off and reading a Javascript book (make sure it's a recent one. Pre version 1.4 books won't cut it - I think the Rhino Book is the best http://oreilly.com/catalog/9780596000486/) and making sure to learn about how to overcome shortcomings of javascript (i.e http://yuiblog.com/blog/2007/06/12/module-pattern/) ...and have a look at jQuery (or Microsoft AJAX library) for example. Because of different browser support issues you'd better off using a library than trying to handle browser support yourself.
3 - XHTML / CSS - This is another must nowadays. I think XHTML is relatively straight forward. Just make sure you know XML (escape your output for example - Security!) and remember, browsers much too merciful. Only a few tricks you need to learn about that one. But CSS is another though nut. Selectors, cascading, floating boxes and so on... add another book to your shopping cart for that too.
4 - HTTP - is the mother of all good and evil on the web. It is a very simple protocol and well worth the time learning it, at least the basics. Headers, redirects, response codes, GETs, POSTs and all the other details are actually very easy to understand and yet I think most of the developers are too busy learning big frameworks, HTTP is forgotten in general. Bonus: Once you understand HTTP a little, MVC will come very natural to you; you will understand why we need state management better; security becomes easier to manage.
5. ASP.Net Web Parts - Web parts is one of the amazing features of ASP.Net which gives the ability to create customized pages. Through this the user can modify the content, appearance and behavior of ASP.NET pages. There are some set of controls in ASP.Net that provides this features. One of the unique features of this is user can drag and drop the gadget sin a page.
6 - Asp.Net as the Framework - Last but not least, Asp.Net is a great framework and I think it is sometimes only perceived as WebForms, which is quite unfair. Asp.Net MVC for example sits on top of Asp.Net but it has not much to do with WebForms. I suggest understanding the life cycle of an Asp.Net Application and grasping the idea of major APIs - HttpContext, HttpHandler and HttpModule for example.
One last thing, but actually very important: Security. I breifly mentioned while I was telling you about HTML and HTTP but there is quite a lot more to learn. I recomend at least getting yourself familiarised with major types of attacks and vulnerabilities. A good place to start is probably http://www.owasp.org/. And keep your security knowledge up-to-date, by subscribing to various security related news feeds.
Good luck.
