I think you meant to ask "do you think I could become a developer" not a designer...for your first site, its a good start. what you need to concentrate on is xhtml, CSS and also start looking in to using JQuery, this is a very good addition to your skill set, however....you will bang your head against a wall for the next few years whilst you get your ahead around being a developer....and it doesnt stop there, you need to decide if you want to be a front end developer or a back end developer. Back end developer would not do any of the pretty stuff, they would write code using ASP.NET, C#, VB, Java etc etc
Front end is the guys who do CSS, XHTML.....
thats just a few pointers for you