I was responding to your statement "It's not actually so trivial. There are simple observations that are consistent with a Round Earth model, but it's pretty difficult to nail things down rigorously and precisely."

I object to the idea that additional rigor and precision is needed for the specific case you were responding to.

As to your overall topic, one step is to watch that miniseries I mentioned earlier, "The Ring of Truth: An Inquiry into How We Know What We Know". It gives several examples of working things out from scratch. For example, drop a measured and small amount of olive oil onto a still pond. It produces an oil patch. The volume of the oil divided by the size of the patch gives an upper bound for the size of the molecules in olive oil.

What you might not know is that there is a large amount of literature in repeating some of the classic experiments. For example, physics undergraduate lab courses may have the students measure the strength of the Earth's local magnetic field, or the value of the gravitational constant G (not g, which is the acceleration due to gravity), and do so using basic equipment you can gather or make yourself.

Detecting the Earth's structure on your own is difficult. You will need a seismometer, and learn how to interpret it. A quick search shows there are many hobby plans for building your own seismometer, or you can buy ones made for schools. The harder part is to come to trust that P-waves and S-waves are different, that they have different speeds, that one cannot go through liquid, and that observations of many earthquakes from around the world can be combined with an understanding of material science and thermodynamics to produce a model of the interior of the Earth.

If your purpose is to avoid what you consider to be appeals to authoritative sources, you will have to redo from scratch what took others several lifetimes to develop. One of these is Francis Birch, https://en.wikipedia.org/wiki/Francis_Birch_(geophysicist) . Quoting that Wikipedia page, "In 1952 he demonstrated that Earth's mantle is chiefly composed of silicate minerals, with an inner and outer core of molten iron."

Concerning the round earth model, what you can do given limited travel is to create a Foucault pendulum, or visit somewhere which has one set up. In principle this can be done in a backyard, but it required a long pendulum wire and still air - do you have an empty corn silo in your backyard? The time it takes to go around can be used to calculate your latitude.

However, this only shows that you are in a rotating reference frame. You then need to repeat what the ancient Greeks did, and look at lunar eclipses to see that the shadow of the earth is always a circle, no matter where the Moon is in the sky.

These together mean that from your backyard you could (in principle) determine that the Earth is a rotating round object.

Concerning your question "Does the moon have an atmosphere?", the problem is, the Moon does have an atmosphere. It's very thin - https://en.wikipedia.org/wiki/Atmosphere_of_the_Moon . A naked eye view of the Moon is enough to determine that it does not have an extremely thick atmosphere. So with a backyard telescope all you can do is set an upper bound for the thickness of the atmosphere.

But to do that, you need some way to predict what those aberrations might look like. For example, if it had a CO2 atmosphere that was 0.1 bar at the surface, would you be able to detect it? Have you measured the index of refraction of different gasses, or are you willing to trust the measurements of others.

I'm not saying it's impossible, but rather that you have to decide at some point where you can trust the results of others.