Replacing an old light switch with a new one. Trying to figure out correct wiring
You are attempting to replace a "normal" switch with a a "3-way" switch, which is why you have a "common" terminal you don't know what to do with.

You also are replacing a switch that depended on the box screws for grounding with a switch that has a ground terminal. You need to maintain the connection to "the ground wire going out the top" and you should also have a "ground screw connected to the box" (unless it' a plastic box - hard to tell from that picture) as well as the "ground screw on the switch" - it is unsafe and improper to connect only to the ground screw on the switch, particularly if (as I read it, which MAY not be what you MEANT) that's at the cost of the ground going out to the fixture. Basically, you need a wire nut for the grounds, and you need "pigtails" (short wires) coming off that to make the connection to the box &amp; switch. "Special for grounding" wirenuts are made that allow one of those pigtails to be from one of the wires going straight through the wirenut - whether you want to bother with buying one for a single switch is up to you, you don't have to use them, a regular wirenut will work.

Edit to add in response to edit of question: Yes, if you have adequate wire to do so, you can just bend the ground wire around to connect to the ground screw on the switch, and keep the wire-nut connection to the top ground wire. The bigger picture does appear to be a plastic box, so you don't have to try to ground that as well.

Best solution would be to get the proper replacement switch. However, you can simply leave one terminal disconnected and use a 3-way as a regular switch:

Can I use a 3-way switch as a single-pole switch?