Short version: I would recommend JQuery UI (A javascript framework), this will make your life easier. As for other components, you are free to stack it up from server-side to client-side components depending on your needs and most of the times depending on your preferences.
[update] The long version: Personally, if I would start a useful website with a back-end, I would use Java as my PL of choice since I'm familiar with it, SQL for my persistence, and make it a Web 2.0 site, like what you want to achieve which is a rich internet application (drag and drop, dynamic content, etc). I would normally make use of Java (JSP, Servlets) but use Grails framework (Groovy, GSP, Groovlets), SQL DB (like MySQL, SqlLite, etc), Ajax, HTML, CSS, Javascript (JQuery as my JS framework). Eclipse would be my IDE to program and integrate all of these components.
Some people would prefer to use Python, others PHP, some .NET, Ruby on Rails, etc. Like what I said, it is very subjective. If you already have a field of expertise, then I suggest you stick with it, but if you are starting from scratch, it's basically a choice of the easiest learning curve for you because different Programming Languages can and will satisfy your needs.
For the front-end you have Flash, Applets / JavaFX, Javascript, etc. The problems with Flash, Applets or other embedded media are 1) heavy on resources 2) compatibility, but the clear advantage is that they are more robust because you can do so much more with these technologies in arguably a smaller amount of time. But if you want to use one of these, you have to seriously think about your target audience e.g. for Flash, iPad and iPhones cannot display them at all (except if you installed the unofficial "frash" which supports a bit of flash content).
I believe what you want to do is highly feasible even if you just use Javascript, and again, I will strongly recommend that you use a framework that can readily satisfy your needs (like drag and drop) so that 1) you'd worry less about cross-browser compatibilities, 2) you won't have to reimplement them and 3) fix many unexpected bugs along the way. But this is the general idea if you are developing a software, if the framework or library already exists, use it (just be extra careful with licensing though but this is another topic) :)