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The American starts in the Sierra slopes west of Tahoe in several forks all draining in to Folsom Lake (20 miles east of Sacramento). There is certainly trout fishing in the upper American system but it's not considered one of the premier fisheries. The lower American is the Tailwater of Folsom Lake and it's forebay Lake Natoma. It joins the Sacramento River (originating from Mount and Lake Shasta to the North) in Old Town Sacramento which all becomes part of the Sacramento Delta joining up with the Bay and out the Golden gate in to the Pacific.

There are Salmon and steelhead runs up the Sac and American. The larger numbers are in the fall/winter run but there is a small spring run of steelhead. Salmon are in rivers in the fall. May is Shad and Striper time in the Sac/American systems. Even during the fall steelhead run, the fish numbers are modest and most anglers pay their dues for few but glorious fish.

The Lower Yuba Drains out of Lake Englebriet an hour and a half or so north of Sacramento and has good trout populations in the tailwaters. It's mostly a drifters section since access is poor and wading is difficult. All of these river systems change character as the growing and dry season starts in May. Flows out of the resevoirs are increased to support agriculture. So get on line for flow rates and conditions. Especially important for Putah Creek which is why I provided the link. It can get completely blown out.

For all of the mentioned river systems including Putah, you will need a CA Licence, Delta Stamp, and Steelhead Report Card.