Since you don’t have any experience on Object oriented programming like C++, (I am assuming this, because you said you have done a basic course in C) the best course to start with will be Core Java. This will give you the familiarity about the language itself. Once you are done with Core Java course(go for Java5), you can take any of the course depending upon your needs. For example, if you want to go for web technologies you can learn JSP, Servlet etc. If you are interested in server side technologies you can go for EJB. You can even go for Web services. But you should get the command over the language by completing the Core java course sincerely.
Core Java and J2ee both are essential. I suggest you to go through SCJP Sun Certified Programmer for Java 6 Study Guide from Kathy Sierra is one of the BEST book and Head First JSP and Servlets for Core Java and J2ee respectively on your own. Even Self study is sufficient. Other than that, If you will go through both books well then you can apply for 2 certification which was previously called SCJP and SCWCD so with 1 proper study schedule you will set eye on 2 Goals.