1996 Saturn SL White smoke from exhaust

Just put in new radiator. Checked antifreeze level a week later, and it is not full but was filled to appropriate level after radiator was installed. Put antifreeze in to bring up to right level.
Checked oil dip stick a week ago, oil level was ok, oil was not black. About one week later, check engine light came on. Checked oil, it was 2 quarts low. Of course, I put in 2 quarts but could not understand as I do not drive past neighborhood area, estimating 30 miles per week. After driving for 10 minutes, engine light went off, that is after I put in the 2 quarts.
Today, as driving out of garage, after car sat for a full day, white smoke came from exhaust area. It billowed into the air, but as I drove a few feet, no more smoke. I go to my destination. Car sits for 7 hours, I start it up, and as I drive off for a few feet, white smoke comes out again. But smoke stops a few feet after that. I touched inside of exhaust pipe and it's black.
What is the problem? Please.