The Home screen keeps up a running feed of your friends' pictures, with options to comment or "favorite" them, similar to your Tumblr or Flickr feed. You can also branch out to look at other pictures from a given user, change your relationship settings, or see what else they've got going on.
The Explore screen (a compass rose) shows you a range of public photos from all over Instagram, in tiny icons rather than full-screen pictures. This is a fun place to start, jumping from one interesting photo and user to another, following the things that strike your fancy.
The News screen (represented by a heart symbol in a word balloon) shows you all the activity of the users you follow: photos they've liked, new feeds they are following, and things like that. You can follow these out, similar to a Facebook Timeline on the mobile app, to see if it's something you'd be into. This is the screen where Instagram's social networking takes place, if you're interested in being a part of that community, and also where you monitor your followers and messages.
The Profile screen (look for the text-message box symbol on the bottom far-right) is the place for changing your settings, profile, looking at your photos and downloading them to your phone, and checking out your photo map. Nuts and bolts, as well as several intuitive menus, make this the most fun screen to play with when you're not actively taking photos or perusing your News stream.
Once you hit "click" on the main screen, you'll have the option to filter the shot, delete it, or automatically upload it to your public or private feed. It is at this point that the connections you made to your other networking services kick in: While the image itself is preserved in the Instagram cloud for later use, the service also sends a one-time push-through of the image to the sites and applications you've specified. Your profile on the app itself -- that is, on your smartphone -- saves your logins and passwords for those sites you've approved, and so the push will seem automatic.
Of course, if there's a breakdown between your phone and the third-party site or app, or in the servers themselves, or if you've changed your password in the interim, you might run into a few complications. It's best to try a few test shots once you've updated your Instagram profile to include those connections, so that you can see how it looks once those images are pushed. If a setting is off, you could have size or orientation issues when it publishes to your other accounts. If Instagram has trouble automatically publishing to a specific network, double-check your logins and passwords to make sure you've set all the app permissions and connections correctly.