This question gives you a hint on how to use Intent to launch a Twitter application and use it to publish. Thus, the user does not need to provide your program with Twitter credentials.
If you want to connect to twitter yourself, you can use one of the java twitter apis that are there.
Hope this is the starting point. If you encounter more specific problems, come back and we will be happy to answer them.
For a general action on Android, check out hello world and all the other tutorials on the Android developer pages.