Depends on your comfort on the roads, desire for speed, shortest distance, etc. You could take PCH to 2nd, left on 2nd, left on Bay Shore and then take the strand bike path west across town. The problem is once you get to the aquarium area it can be hard to find the LA river trail if you've never been there before. Technically you could take PCH all the way to the LA river, but there are a number of not so great areas you would have to ride through, including the traffic circle. You could also take Willow from the San Gabriel River trail west across town to the LA River. Off the top of my head I recall the LA river has on/off ramps at Anaheim, PCH, Willow, Wardlow, Del Amo, as well as some others.
Or just let some of us locals know when you'll be riding through and maybe we can meet and show you the way through town.
This map might help you understand how some of the paths interconnect.
http://www.metro.net/riding_metro/ma...a_bike_map.pdf