What device are you on? Doesn't really matter the basic procedure.
1. Click the "app drawer", or the "all apps" menu (If on HTC, on the main screen, an arrow symbol pointing upwards on the low-left corner
2. Scroll & find "Market"
3. Click market
4. Find an app that's interesting
5. For example, search with "planet"
6. Click one of the apps that come up
7. If the app you have chosen is free, then there is a button on your right that says "free".
8. Press the "free" button, it should turn to "install"
9. Press install (you should see a message; your app will be downloaded or something like that)
10. Wait for the app to load (If on HTC, drag down the main top-bar to see it's proggress
11. When ready, repeat steps 1 and 2, searching for your new app
Hope this helps some!