Here are four music services that you can listen to, for free. You can use them any Internet device you have, or all of them: your smartphone, tablet, and/or computer (including Chromebook). Since they're streaming services, they don't download and store music on your device, so you don't have to worry that they'll use up your space.
Pandora is available on everything -- as an app for your phone or tablet from your App Store, as a channel in smart TVs and streaming boxes (like Roku and Apple TV), or as a web app or web page on computers and Chromebooks. Pandora lets you enter an artist's name, or a song, or even a genre like "Show tunes" as a seed, and it creates a "station" that plays the artist and/or song along with similar music.
Pandora lets you create up to 100 "stations" and you can refine them by adding artists to a station, and by using the thumbs up/down to help Pandora know what you like. Once you've created your free account, Pandora will give you your stations on any device you use, so you can get your stations on your smartphone as well as on your computer and/or tablet. If you want to go ad-free, or be able to skip songs more than 12 times in a 24-hour period, you also can subscribe for $4.99 a month.
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This is one of my new favorite services, and it's also available as an app for iOS or Android, as a web app for Chrome, or in the browser on your computer. Unlike Pandora, Songza has pre-made, curated "channels" (aka sets) which you can browse as genres, moods, activities, or decades. Each of these sets will have from 3 to more than a dozen playlists, and each of those playlists will have 50+ songs.
Songza also has a "Concierge" that will suggest music for the time of day, and offer a half-dozen or so categories in each selection, with three playlists in category. One thing I like about Songza is that I hear more and different than in my Pandora channels. Songza has just been bought by Google and is free and ad-supported, but the ads are just a pre-roll video before the channel starts, and an ad banner at the bottom of the app. If you want to get rid of those and have more skips, Club Songza costs 99 cents per week ($3.99/mo).
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Don't let the name fool you -- Slacker Radio has music that will please everyone, with curated "stations" by celebrity hosts, and also has non-music selections like ABC News, ESPN, and Weather. Like Pandora, you can create your own stations (although not at the free level), and you can find music by genre, decades, artists, and radio-style "Countdowns" like "55 Top Songs About Summer."
On the free level, you can't create stations or playlists, but their selection is quite rich. They also have $3.99/mo and $9.99/mo subscriptions that let you go ad-free and do more.
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Spotify works on mobile devices with an app, and on computers with a downloadable app for Windows and Mac, and a web app for Chromebook. It's also just been launched in Vizio Smart TVs. The free level lets you choose from their pre-set "shuffle" playlists and is ad-supported, while the premium level ($9.99 a month) allows you to choose any song you like to play, and also lets you download and listen offline. Spotify does not have a free level on the smart TV.
Spotify's differentiation is their "music discovery" and social sharing of music and playlists with your friends who are also Spotify members. (I don't think I have any, lol.) Spotify also has developers who create extensions to build special playlists and such things. This makes it something with lots of surprises when you poke around, but not my first choice.