When I flush my upstairs toilet, the downstairs toilet floods and then slowly drains. What's happening?
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.