Android Phones
If you have your phone's own sync
software installed on your computer, that may be the simplest way to
sync your playlists and music.
You may be able to sync with Windows Media Player instead, if you don't want to install your phone's software or just prefer WMP.
Alternative Method - Directory Sync
On some phones & players, you can sync your player by copying all the contents of your computer's "My Music" folder (songs and playlists) to the player's "MUSIC" folder (or use a directory-sync program such as freeware DirSyncPro to do the equivalent). If you can do an ordinary Media Sync with WMP or your phone's sync app, that's probably easier. If you do a Directory Sync, you must have the option checked for Write songs as relative paths in Jukebox Shuffle options. First save the playlist in the default .WPL format, which works with all the Android phones I've tried. If WPL doesn't work with your phone, try M3U (Unicode UTF-8) format as the most likely alternative.
Windows Phones
I presume Microsoft has made sure that syncing with Windows Media Player will work easily with Windows Phones. That was true with my Windows phone years ago, but I don't have experience with recent Windows phones.
MP3 Players
This is simple for most MP3 Players (aka Digital Audio Players), because almost all except iPods are designed to sync with Windows Media Player. Just use the USB cable that comes with your player to connect it to your computer, then start WMP and click its "Sync" button at the top of the window. It should detect your player. In the "Windows Media Player - Device Setup" window, make sure the shuffled playlist you created is in the "Playlists to Sync" list on the right side of the window.
iPods & iPhones
If you have an iPod/iPhone, you probably
already have iTunes installed for syncing your player. When you save the
shuffled playlist you should select the M3U (Unicode UTF-8) format, which iTunes
recognizes (at least according to what I've read on the web). Then in iTunes,
tell it to include that playlist in the sync.
Online I've read about Windows Media Player plugins that enable it to sync
with iPods and iPhones. The ones I've seen aren't free, I haven't tried them, and
I don't know whether it's legally allowed by Apple Corporation. Whether to try
them or not is up to you.