I use to live in Antioch and for what you are asking for is not to far away. Take 580 (I think that is the freeway to go towards Livermore) and when you hit I-5 Go south. There is an area to go wheelin about 5-10 miles down I-5. I don't remember the offramp but I'm sure if you poke around you will find it. Mostly flat, river beds, had some great mud lots of open space.
You can also cruise down to Holister Hills. Take 101 to 25 go to holister and follow the signs. Really great off roading there.
Have fun