If you signed out in the app (not the web site, it doesn't matter if you are signed on the web site or not) the next thing that happens is the app takes you to a sign in screen. If that isn't happening, try exiting the app completely. (click the home button twice, press and hold an icon on the bottom of the screen until the red x shows up, then find the tripit icon by sliding right or left there and press the red x, then press the home button once, pause, and once more to get back to the home screen.) Then try again.

if you have an up to date iphone, you should be able to reinstall apps from the app store applet. After you remove the app, open the app store and go to updates, then purchased.

-David