Recently, the Seattle Department of Transportation installed a 2-way bike lane about 2/3 of the way around Green Lake and extending beyond. In the coming years, they plan to complete a full bike network around the outer loop of the lake. These bike lanes are great for getting around (especially to geocaches) which you may find useful here. Good luck and enjoy the lake!
As a note, The hint contains a spoiler, so only use it if you need it!