Welcome to these forums! They have proven to be a huge source of information and education re: my iPad and I hope you'll find them to be just as helpful. I'm so glad you're enjoying your iPad. I see there aren't any responses yet to your thread, and I'm no pro, but I will tell you what I can.
You will not be able to download just anything directly from any source on the internet, even if you jailbreak. Your ipad will always use "apps" that are available at iTunes, and for jailbroken ipads, from Cydia. (Oh, and you don't have to sync through the desktop/iTunes exclusively if your iPad is using the current version of the iOS software... you can do that now through iCloud via your wifi connection.)
So far, I haven't jailbroken my iPad, only because I've been satisfied with the apps and functionality I get in its "native" state. Many, many users love the options offered by jailbreaking and are eagerly awaiting the newest jailbreak which is said to be coming very soon.
You can learn more about jailbreaking and what it offers by visiting the "iPad Hacking" section of these forums (The section on the main index just below this one!
http://www.ipadforums.net/ipad-hacking/ ). (And please know, in this instance, "hacking" is not used in the potentially negative context you may be accustomed to!)