I logged out of the iCloud account on both my iPhone and my iPad so neither devices sync each others music or apps anymore, but whenever I make a note on my iPhone I always find it on my iPad and I do not want it there. Is there a way to make my notes stay only on my iPhone?
Notes may sync via other email services, like gmail.
You need to check the settings of EACH of your email accounts, and, if necessary, disable syncing notes.
It is also possible that notes sync to your Mac when you connect your iPhone to iTunes (via the USB cable or wireless). To avoid this, you might want to check iTunes sync settings (not 100% sure about the latter is still valid, it was definitely present in older versions of iTunes).
You could also try using one of the many note taking alternatives on the app store - the free Note app from Squarespace is my favourite.