Like my long title says, downstairs toilet is flooding after either of my upstairs one flush but then it slowly drains. It drains to where there is almost no water in the base, past the normal point. The downstairs' toilet's tank also has bubbles rise up from some black cylinder part. I am obviously not a plumbing expert and just looking for advice. A family member says we have to snake the toilet, is that what we should do?
Tyson's response in comment is spot on. Without knowing your house layout, you can follow the drain from the downstairs toilet to where it meets the pipe for the drain from the upstairs toilet.
Once you identify the drains intersection, look for another drain from any sink anywhere in the home that is PAST the intersection of the two toilets pipes AND closer to the main sewer drain.
If there is a sink, run water in it full, drain it, time it. Do it again, but this time flush the toilet. See if the water drains slow. This is just a simple test if do-able to "guess" whether there is a plug/buildup in the pipe where the toilets meet. Problem areas often will be at bends, turns, or where pipes are joined together. If there is buildup on the outside of the pipe, it could be a sign of buildup inside too.
There is even a possibility your issue is tree roots or broken sewer out to the main - it can be a pricey job and will only get worse.
If you can snake it, and are alright with opening the clean outs, go for it. Otherwise call a plumber. And be aware - if you were to snake haphazardly, you can do more damage than may already exist, especially if the pipes are old/brittle.
If you have a septic system that has a clog it will back up at the lowest point in home. If it's not a full tank find the pipe they pump out of and pop the cap. A broom handle is generally long enough to reach the junction. Try to work the obstruction free and Just listen for water flow and the clog should be cleared.