How does packet switching work, exactly? (Especially over wireless?)

I understand that packet switching involves chopping up a file into packets and then sending each packet along whatever route is the best through the Internet, in any order, then reassembling them at their destination.

So, let’s say I take a picture of my cat on my phone and e-mail it to my friend. How exactly do the packets move? Would it be:

1. Phone chops up file, and the packets all move to a local cell phone tower.

2. Packets move willy-nilly to e-mail server at some giant data center.

3. Packets move from there willy-nilly to my friend’s computer, where they are reassembled.

? (In particular, I’m confused as to how they flow over wireless. Also, how many packets does a large image file usually get cut up into?)