BuyFromEU

For using OSM on Android:

• I use OsmAnd, which is free on F-Droid. It's very feature-rich, and I can customize it pretty much exactly how I like. There's a good tutorial for beginners here.
• However, some people prefer the simplicity of the app CoMaps, which is available for Android and iOS. It's a fork of Organic Maps that arose over disagreements with Organic's governance (see: open letter).
• There are also specialty apps like Alpi Maps for more specialized uses of OSM (in Alpi Maps' case, hiking).

For contributing to OSM on Android:

• I use Vespucci for more complex edits on Android, so things like adding features. (The rough iOS equivalent is Go Map!!.)
• I use StreetComplete for a more trimmed-down, gamified experience. It isn't a power user experience like Vespucci, but it's often really useful if you're just on the go and want to make small but meaningful contributions.

The easiest way I've found to contribute local information is the iD editor on desktop; it's the one that pops up if you click 'Edit' on OpenStreetMap.org. It strikes a good balance between user-friendliness and power user stuff. For full-on power users, JOSM is a good option. And the easiest way to contribute in a limited way is StreetComplete, which will ask you questions about existing features (like "does this traffic signal have a walk button for pedestrians?")

Edit: I linked to but forgot to specifically mention the OpenStreetMap Wiki, which is a great general resource.