I admit it, my gut reaction to this question was “jeez, this isn’t that hard. just start up Skype for iPhone and it shows you how to do it” but that’s not so helpful, is it, so…
You’re right that the fastest and easiest way to download an application onto your iPhone is to tap on the “App Store” icon, find the app and download it directly, without involving iTunes or anything else. I’ve another article that details how to do just that, but for now, let’s assume you know how to do it and you’ve successfully copied Skype for iPhone onto your phone.
Now tap on the Skype icon and you’ll see the beautiful splash screen:

After a few seconds you’ll see:

Let’s start by going down the path of you not having an account on the eBay owned Skype service. You can set up one by tapping on ‘Create Account’ on the bottom of the screen.
Hold on, though, you can’t use Skype as an emergency 9-1-1 service, and you’ll have to accept that limitation before you can proceed:

Finally, it takes you to a spot where you can enter a variety of personal information and set up an account, easily enough:

I already have an account so instead of signing up for another one, I’ll go back to the home screen of the Skype app and enter my login credentials:

Once I’ve done so, I’ll tap on the “Sign In” button on the top right and…

(I’m trying to build up some suspense. Is it working?)
After a few seconds I learn that I have entered my credentials properly and you get a first meaningful look at the Skype for iPhone application:

Very nice, a list of my contacts who are on the Skype system at that moment. To contact any of them I simply tap on their name.
Easy and a very attractive iPhone 3G application to boot.
Nice job, Skype. Now to try out the application itself…