I want to create a notation for 2 different notes being played at the same time (half and quarter notes) in which the half note is played at first and the quarter is played at the second half of that half note. Is it possible? or the quarter note just cancels out the half note?
You can use something like the arc left of the legato word, going from one note to the same note, while having different notes on top (in this case).
So, in your case assume you have a C as half note, and the E quarter notes starts later, you draw two C quarter notes, connected with an arc, and above the second C you draw the E quarter note.
It's actually quite simple and pretty common. It is called a split voicing, in which two (or more) parts (or voices) are written on the same staff, allowing them to differ in many ways (such as rhythm). In most cases (particularly when both parts will be played by the same person, as in the piano), you can add and remove voices at any time (like for a single measure), so you don't have to write the entire piece with everything split.