A lot of people seem to be moving to web applications these days; the good news is that it isn't as hard as you might think. ASP.NET takes advantage of a lot of the same design patterns for the UI framework as WinForms, so it should be pretty straightforward.
Also, moving to ASP.NET, you can still leverage all your knowledge about C#, since it's one of the supported languages for the technology. You'll be up and running in no time.
There are a couple ways you can build user interfaces with ASP.NET, but the most common is ASP.NET WebForms. You can also use frameworks that provide some abstraction like ASP.NET MVC (similar to Ruby on Rails) if you want to separate your business logic and your interface a bit more.
I'd suggest taking a look at http://www.asp.net, which has a ton of great resources for getting off the ground as well as videos, samples, and walkthroughs. Best of luck; I'm sure you'll be fine!