When I started learning piano everyone kept telling me to play "Mary Had a Little Lamb" and I wanted to throw my keyboard out the window. No offense to whoever wrote that song but I didn't buy a piano to sound like a kindergarten recital.
Took me a while to find songs that were actually beginner-level but didn't make me cringe. Songs people would recognize as actual music. Here are the ones that worked for me.
Lean on Me by Bill Withers. This was the first real song I learned all the way through. Simple chord progression – C, E minor, F, G. Slow tempo. The melody stays mostly in one position. Sounds like actual music even when simplified. Took me about three weeks to get passable. The chord theory behind it is covered in my guide to understanding chords.
Let It Be by The Beatles. Four chords: C, G, Am, F. The famous I-V-vi-IV progression that powers like half of pop music. Slow enough that you have time to think. Everyone knows it so they're impressed even when you play a basic version.
Hallelujah by Leonard Cohen. Arpeggiated chords in the left hand, simple melody in the right. The pattern repeats so once you learn one verse you basically know the whole thing. Beautiful song that sounds way harder than it is.
Imagine by John Lennon. That iconic piano intro is actually pretty beginner-friendly. Left hand does broken chords (arpeggios), right hand plays melody. Good for building the left hand independence you'll need for more complex stuff.
A Thousand Years by Christina Perri. The Twilight song that was at every wedding for a decade. Slow tempo, simple progression. My friend who started at 40 learned this as her first impressive piece.
River Flows in You by Yiruma. On the harder end of beginner but doable after a few months. The arpeggiated pattern in left hand is beautiful and not too complex. I failed at this at month two, came back at month six and got it.
Comptine d'un Autre Ete from Amelie. Sounds impressive, mostly repetitive patterns. The left hand has a specific figure that repeats while right hand plays melody over it. Great for showing off.
Where to find beginner arrangements: MuseScore has free user-uploaded versions. Musicnotes has quality paid arrangements labeled by difficulty. Search "[song name] easy piano" and look for anything marked beginner or early intermediate.
My approach to learning new songs: Listen first, get it in your head. Right hand alone until solid. Left hand alone until solid. Combine stupidly slow. I cover this in more detail in my complete guide to learning your first songs.
Don't start with your dream songs. Build up to them. The songs above are stepping stones that sound good while teaching fundamentals. Save the Chopin for later.