Why do my pedals move when I move my wheel in both directions? How do I fix it?

This bike definitely has a derailleur and is not intended to operate as a fixie (i.e. something is definitely broken.)

A normally functioning multi-speed bicycle has a mechanism that allows it to pedal forward and coast (freewheel) when standing on the pedals. Many older or cheaper bicycles have a "freewheel" which accomplishes this. Other bicycles have a freewheel integrated into the hub body, a "freehub." They're essentially the same mechanism - a ratchet with a few pawls.

In all likelihood, the pawls inside of the freehub or inside of the freewheel have seized or have come apart. Procure a can of penetrating oil and spray the seam where the cogs and the hub meet (i.e. where it is supposed to rotate. I've included a photo with arrows indicating where to spray. Be careful not to get oil on your rims or brake pads.)

If this doesn't loosen it up, you'll need to either replace the freewheel or overhaul/replace your rear hub.

If you flip the bike upside down (as you have done), then "pedal" in the "right" direction (as if you were trying to make the bike move forward), then let go of the pedal, the pedal should stay stationary (or maybe drift around slowly) and the chain should remain relatively straight and taut. If, when you let go, the pedal keeps moving pretty much at full speed or the chain bunches up, then the freehub/freewheel is not "free".

If that is the case, the first thing to check is for any sort of trash (leaves, string, etc) caught between the large rear sprocket and the hub. If you find nothing then the rear wheel needs to be serviced.

Since your mechanical ability is apparently quite limited, I'd suggest you take the bike to a shop, rather than attempt to fix it yourself.