Notification Center isn't just for your lockscreen. You can also access it at any time.
To do this, swipe from the top of the screen towards the bottom. This will pull down the Notification Center screen (it can sometimes take a few tries to get the hang of exactly how to swipe to make it appear, but keep trying. Once you get it, it'll become second nature).
The Notification Center pull-down contains alerts from all the apps you've configured in the last step to appear here. You'll see your local weather (swipe it to the left or right to get a six-day forecast) and other apps.
For the other apps, if you turned Show Preview on in the last step, you'll see a little bit of text for each item. Tap on an item to jump right to it in that app. If your alerts are longer than your screen, just swipe up and down within the screen as you would when scrolling through any other content.
To hide the Notification Center pull-down, just reverse the swipe gesture: swipe from the bottom of the screen to the top.