Be sure that you are in the "Songwriting" mode of the app.
You don't actually need to save your song, as the app will save it automatically. If a new song is opened, you will be asked if you want to save your song or the changes to your song.
If you want to save the song anyways, be sure that you are in the "Songwriting" mode of the app, then swipe up the sheet at the bottom of the screen and tap on the icon highlighted below.
If you don't see it, you must scroll the list of icons horizontally.
To change the name, make sure you have the latest app version. Open the song, then tap on "save" (see above). In the following overlay, you can change the song name and tap "Save".
You will find your songs in the sidebar menu and "Saved songs".