This is a very simple experiment that only requires your naked eyes and a sketch book. (But a camera or cell phone will help). 
Showing varying geometrical shapes of constellations is necessary proof for a Flat Earth.

This is a more technical experiment that requires a camera, preferably a digital camera capable of capturing raw images. Using raw images is very common when doing this kind of work since it eliminates all discussion about possible forgery and "photoshopping".  The internet (YouTube etc.) is inundated with various video sequences of solar movement in the sky, sunsets, etc. Quite often is this material rather weak. Doing this experiment gives you the ability to provide much stronger evidence.
Showing varying sizes of the solar disk is necessary proof for a Flat Earth.

This is the most famous astronomy experiment of all. But this one requires teamwork with other observers located on other places.
The experiment will support the geometrical model which fits the experimental data best.
Fitting the data correctly can produce sufficient proof for a Flat Earth.

Bonus experiment. Celestial Navigation

This is by far the most difficult challenge. You will need knowlegde of college-level maths such as linear algebra, vector operations etc. Set theory and combinatorics would also help. Knowledge in programming will be very helpful too. Expect several weeks of hard work.
A successful method for Celestial Navigation using a Flat Earth model will produce sufficient proof for a Flat Earth. 

Some useful tools.

For celestial navigation, you can pick up the complete software kit, as documented here. It contains quite a lot of documentation and a Python library. It can be used on a cellphone without an internet connection.

The same software kit is available in this web application (a Google Colab notebook). You can enter parameters from sextant, chronometer, and nautical almanac readings and use them to find your position.

You can also check out this simple curvature calculator which is useful for finding the obscurations, horizon drops, and distance of line-of-sight.