I-75/the Chrysler Freeway is located directly east of Royal Oak, and runs north-south. I-696/the Reuther Freeway is located directly south or Royal Oak, and runs east-west. Bordering the left-hand side of Royal Oak is Woodward Avenue, which runs northwest-southeast, and will take you southbound directly into Detroit.
By train
Amtrak
201 S Sherman Dr, (http://www.amtrak.com). Train service to and from Chicago on the Wolverine Service 5-6 hours, $25-$50, with many connections in Chicago. Deeply discounted tickets at short notice are often available at Amtrak's Weekly Specials page (http://tickets.amtrak.com...).