The starter was smoking so we decided to replace it. By the time we did replace it the battery had died. When the jumper cables was attached it started itself up without the key being put in the

Sounds like your starter relay is sticking!. Change your starter relay in fuse panel- cheaper and easier and that does not work move to your ignition switch that could be faulty.

What year cobalt? These are known for ignition switch issues. Check yours to see if its in the recalls or service bulletins. Failure to fix the ignitions on these vehicles could result in serious injury or death.