As you said, all the notes "on the beats" in the first 2 bars belong to a C major chord, and the piece ends on a C major chord. So it's a reasonable assumption that the key is C major.
Before you can identify the "chord changes" you need to decide on the "harmonic rhythm", i.e. where the changes occur.
The first step towards that is finding where the cadences (at the ends of the phrases) are. There seem to be two 4-bar phrases. That would suggest some sort of cadence leading away from C at bar 4, and back to C at bar 8.
In bars 3 and 4, the notes on the strong beats are D E F G which suggests using a standard way to harmonize a scale ending on the dominant chord (G).
The second phrase, bars 4-8, again starts with two bars of C major chord, and the last bar looks like a perfect cadence G (or G7) C, taking the right hand C at the start of the bar as an unessential note. Copying the harmonic rhythm of the first phrase, you need two chords to harmonize bar 7.
Of course you could do something completely different - e.g. take bar 4 as a nearly-but-not-quite-Phrygian-mode cadence like B-half-diminished-7th-second-inversion E-minor, and then fit everything else around that idea (which would most likely not involve starting and ending on a C major chord) .....
But that's the difference between "getting an A grade in a Music Theory 101 exam by demonstrating that you know how to apply the conventional rules of common practice harmony and give what examiners expect to see as the right answer" and "being a composer." (Look at the first volumes of Bartok's "Mikrokosmos", for more examples of what I mean by that.)