There is a wonderful intersection of Physics, Mathematics, Applied Math (numerical analysis) and Observational Astronomy in a systematic analysis of the movement of four bright Galilean moons of Jupiter!
They are easy to see in a small telescope, they have well defined orbits around Jupiter, and they have some trick up their sleeves that you can analyze.
Here's an exercise you can do right now. When Jupiter is anywhere near quadrature, it's shadow extends behind it by an angle of almost 11 degrees, so the moons will often be eclipsed. That means instead of appearing or disappearing into Jupiter's overwhelming brightness, they can "pop" in or out of visibility as they pass out of or into Jupiter's shadow.
Jupiter radius 71,492
You can predict when these happen, and then watch to see them and confirm your calculations. You can start timing them to get more accurate information.
From a mathematical perspective you can check the periods of their orbits (from your measurements, or from a table) to see if there are any surprising numerical relationships!
You can check the inclination of their orbital plane to see if the eclipses always happen or sometimes don't.
You can read the answers to When will the next series of mutual eclipses of Jupiter's moons begin? and start planning for 2021 when these begin again!
You can learn to do numerical integration of solar system bodies to simulate their orbits, or use the Python package Skyfield or the JPL Horizons website for high accuracy positional information.
These can also help you calculate the eclipse times by Jupiter, and the mutual eclipses of Jupiter's moons.
Finally, you can try to measure the speed of light by seeing that the eclipses come sooner or later than average when Jupiter is closer or farther than average from the Earth. This is one of the early ways that the finite speed of light was verified.