You can probably accomplish all you want with Flash, though in a practical sense you are probably better off pursuing other web technologies since you would be limited to the dimensions of the Flash stage. Most ecommerce aite would avoid Flash for product displays due to the large quantities of items offered. If you would only have a few items for sale then Flash is a bit more reasonable to consider. In any case, you are more likely to get good information via taking a course in ecommerce design or finding a forum that focuses on designing for that.
To learn how to make Flash interact with databases, here's a link to a tutorial... http://www.gotoandlearn.com/play?id=20 If you go to the main page of that site (http://www.gotoandlearn.com) you'll find numerous tutorials that may be useful for what you might pursue.
You call yourself a designer but if you're very interested in databases and shopping carts you might be more of a web developer or maybe a designer-developer. gotoandlearn that Ned mentioned has some very good tutorials aimed at accomplishing specific task and it's free. In my opinion the best web design and development learning instution around is lynda.com they have around 40,000 tutorials mostly on everything Adobe. It cost $25 a month. If you want to learn about databases you may as well take a class in middleware you'll learn all you need to know about databases while studying middlewhere. PHP is the hot middlewhere now you can learn that or if you want something a little lsss challenging you could try coldfusion. If you want to be a master flasher then you should master as3 it takes a little getting used to but it's awsome. While you're at it you might want to learn Flex. Of course on the design side it's nice to be real good with photoshop and maybe illustrator etc.