The next thing you notice--if you are far enough north--is that the days aren't the same length all the time. At 5 degrees north or south, the difference is about an hour. That may or may not be very noticeable, but by 10 degrees it is 2 hours and I think that's enough of a difference that you can tell. The Sun isn't always in the same position when it sets, though if you live in a forest you might not be able to see that. If you live in the mountains it can be very obvious that sunset is at a different place on the distant slope than it was last month.
If you live at the equator, unless someone has given you a reason to look, I don't think you'd spontaneously notice the changes with the sun. The planets are another matter, of course.