Although you might be able to find some free sheet music on the web-- you could start with
https://musescore.com/-- most sheet music sites will want you to pay for the songs you download, so in the long run it will be cheaper per song to buy a book of sheet music rather than buying a song at a time.
If you're interested in learning popular songs, I believe you'll want to search for "lead sheet," which is a version of a song that's usually written as a simple melody, possibly with chord names written above the melody.
And if you want to get an entire book of lead sheets, you'll want to search for "fake book," which is a sheet music book containing nothing but lead sheets. There are fake books available for different musical genres-- pop, rock, jazz, movie soundtracks, show tunes, etc.
EDIT: However, if you're learning to play the piano then it's good to use sheet music books that are written specifically to be used as lessons, because they'll contain songs and exercises designed to help you learn the different scales, master fingering, come to grips with different rhythms, etc. But that doesn't mean you can't work on learning tunes that you like at the same time, because it can be boring to focus exclusively on scales and exercises-- throwing in a popular song or two that you're familiar with can help give you something to work towards.