You have two choices. Number one is easier. Number two is more difficult.
Number one just continues where you left off.
After you get a requestToken, you will need to launch a WebView and point the URL at requestToken.getAuthorizationURL(). The user will then log in and choose whether or not to allow access to his/her account. Next, if he/she hit allow, an access code will be displayed the user must copy/paste in your your own application. You will use that key with getOAuthAccessToken() (I think, I used the difficult way described later) to get the auth token that you should store somewhere permanently. At this point you are authenticated.
Number two also continues where you left off minus one detail...
twitter.getOAuthRequestToken(REDIRECT_URL).
That redirect_url must be set inside your twitter developer account first. Then follow the same steps as number one except that your webview needs to be customized. You need to use setWebViewClient() on your WebView and create a new class that extends WebViewClient. Inside the WebViewClient's onPageStarted, check if the URL starts with your return url. And then get the oauth info:
String oauth_token = uri.getQueryParameter("oauth_token");
String oauth_verifier = uri.getQueryParameter("oauth_verifier");
Use the oauth_verifier with twitter.getOAuthAccessToken() to get your token.