Search in the browser
Open Book Search Party, search a title or author, and keep the free-only switch on. The result page lists the free sources and a download link when the source provides one.
Stream or save
LibriVox pages usually offer streaming and MP3 downloads. Internet Archive items can be streamed in the browser or downloaded. Save files to your phone's audio app if you want offline listening.
Use your existing player
You do not need a special audiobook app for MP3 files. Your phone's music or file player can resume most downloads. For chaptered books, check the source page for a zip or playlist.
Watch the data
A full novel can be hundreds of megabytes. Download on Wi-Fi, and keep one short title cached for gaps in coverage.