The Lower Klamath is a great river for introductory level class II-III. Most people put-in at the mouth of Indian Creek, which is located in Happy Camp, CA (Home of Big Foot). Indian Creek itself has been kayaked and rafted although you would need significant flows. Another great tributary to check out is Elk Creek, which can be observed by going over the Klamath at the bridge just upstream of the mouth of Indian Creek.
If you are planning on running the 18 mile stretch down to Coon Creek or Green Riffle, you most likely are going to have to shuttle your car(s), which is a simple task since the road runs along the river. If you only have one car and want to save some cash, you can bike the shuttle in about two hours or less if you have a road bike.