I'm not sure you'll find a single app that will get you everything you want.
The most comprehensive app is probably Google Maps. It will have most trains, many local public networks (but by far, not at all of them), and often also includes ride-sharing services (Uber and equivalents), as well as many flights.
But in many places the only option to get local transport info is the local network's app (if there is one) or their website.
Even for long-distance travel (especially trains), you'll usually need the national or regional operator's app or website to buy tickets (and sometime multiple apps).
If you have to travel complex train itineraries, special mention for the SBB app (and website), especially if you travel to/from/through France: SNCF apps are incapable of producing an itinerary with more than 2 stops, while the SBB app will give you those itineraries (you can then split it in 3-leg trips for separate booking).