Moovit (free) is a transit app for trip planning, live arrival info, schedules, maps, service alerts and more. It supports more than 50 cities in the US and many major cities around the world. The app covers busses, trains and Metro Subways.
The app can use GPS to determine your current location, then give you a look at what transportation options are available. Click on a transit station and you'll see real time arrival times. The app includes walking directions, and alerts if there are service disruptions. You can see those in the app, or set up notifications through iOS.
Moovit provides a step-by-step route to get you to your designation, including walking directions. It's pretty easy to figure out, and while it told me that there aren't any bus or train routes in my section of southern Arizona, some nearby big cities offers several options. The trip planning functions worked well. I also looked at London and Brussels and found good up to date information as well.