A chord is not the same as a single note -- it is an abstraction that represents a combination of multiple notes that sound good together, including those that can be used in the accompaniment parts. In this case, you have a C chord, which represents any combination of C's, E's, and G's.
So in your first two measures, you have the notes
| E E E | C D E |. The chord symbol 'C' means that these notes can be harmonized with any other C's, E's, and G's.
Incidentally, you'll notice that one of the melody notes (D) doesn't even appear in the chord. When this happens, it's called a "non-chord tone" or a "non-harmonic tone", in this specific case, since it's part of a scale between two other chord tones, it's also called a "passing tone".