I didn't really know how to answer this without seeming really condescending. So I apologise in advance, but you did ask...
In normal operation, when a message comes in, you pull the notification bar down like you would normally, and select the text message, like you would normally, and Handcent will load and display your message, just like the standard Messaging app does. It's all exactly the same only you'll be using Handcent instead of Messaging.
If you want to write a text message, then you just open the app drawer (where you see a list of all your apps), and press your finger on Handcent, in exactly the same way you would run any other app on your phone. There's no difference. Once Handcent has loaded, press on the little icon at the top left that looks like a piece of paper with a pen writing on it (that's a big clue) to write a message. Alternatively, to read a message from this screen, just press your finger on the message thread you want to read - again, exactly the same as the standard Messaging app.
There are a few of things you may want to do to make this whole process easier. Firstly, when a message comes in the phone won't know which SMS application you want to use so when you get a SMS message and pull the notification bar down and select the message, it will ask you which app you want to use (Handcent, or the default one). When this happens you want to check the box that says "Always use this app" or "Make default" or whatever it says, and press on Handcent. From then on it'll default to reading messages in Handcent.
You may also want to make a shortcut to Handcent on your home screen. Handcent has a widget, which is better than a plain shortcut, so you want to use that. Long press on a blank space on your home screen, select Widgets, and find Handcent, and add it. You can then see how many unread messages you have on the Widget. If you press on the widget it'll load Handcent - brilliant eh?
Finally, you want to turn off notifications in the standard Messaging app, otherwise you will get TWO notifications in the notification bar (one from each app) every time a text comes in. You may also get two lots of beeps. To do this, run the Messaging app (find it in the app drawer like before), go to Menu, Settings, and uncheck the Notifications option.
Done!