Hi,
There's lots you could do. Some ideas:
1. Using an inclinometer app, measure the elevation angle of the pole star (Polaris) and note how it is almost exactly the same as the latitude reported by your phone (the compass app on an iPhone gives you a quick lat/long readout, or just use one of the mapping applications). Get a friend who lives a reasonable distance away from you, ideally north or south, to do the same exercise. Try to calculate the distance to Polaris by treating the earth as flat and using two observations to triangulate, then observe that this doesn't work for more than two observations - you get different results. Then try it with a round earth, and note that it absolutely does make sense for all latitudes in the northern hemisphere (you can't see it from the Southern Hemisphere...why?) if you assume the distance to Polaris is much larger than the earth's diameter (ie your observation lines are essentially parallel).
2. Whilst undertaking the first task, take some time to do some time lapse photography - again very possible with a mobile phone - and get your distant friend(s) to do the same. Try exposures of different times, and note that the stars blur into streaks after exposures of more than a minute or two. Note also that the stars appear to be rotating around the pole star in a circular motion. Observe the same type of photo from a different latitude, and notice that the stars are doing the same thing, ie rotating in a circle, but the centre of rotation is just higher or lower in the sky.
FE theory seems to try to explain (1) by invoking EA, or 'bendy light', but if this was actually occurring, ask yourself how the stars could move in perfect circles, observed from any location, without distorting due to EA.
3. Measure the angle travelled by any star in one of your time lapse photos compared to the time of exposure - the rate of movement will equal the earth's rotation rate.
4. If you're feeling a bit more adventurous, get flying. Get a friend to fly to Australia, while you travel to Africa. Pick locations where it is dark at the same time (check
https://www.timeanddate.com/worldclock/sunearth.html) and take some photos of the night sky at the same time as each other. Notice how the stars you observe are the same as each other, other than differences caused by your respective latitudes. Each take a bearing using your phone to a particular star or constellation - I'd suggest true bearings, rather than magnetic, to avoid complication. Ask yourself how, if any of the FE maps are to be believed, you could both be looking at the same thing at the same time whilst facing in different directions. This only works if the earth is globe shaped.
5. While you're at it, each of you take a flight from your respective continents to the other continent, at the same time. Record the time of flight, from the moment you take off, to the moment you land. Look at any of the FE maps and ask how such a journey could be possible in a subsonic aircraft if the distances indicated by the map are true. And because you've done the journey both ways at roughly the same time, you know it can't be due to 'anomalous winds'.
Just a few to get started with - there are many others. Hope that helps