I think the important thing to remember is that climate change doesn't inherently *cause* anything. This isn't a disaster movie where climate change will suddenly cause the apocalypse. It's a lot more gradual process, which is partly why it's such a hard issue to get people to care about.
What climate change does do, is accentuate weather patterns. Summers become hotter, winters become harsher, droughts (which California was already prone to) become more intense, hurricanes become more common and/or powerful, and so on and so forth. It's not going to be some super-storm or heat-wave caused by global warming that winds up killing us, it's more than likely going to be crop failure. If the climate changes enough to where major crops begin to fail (corn especially for the US), you're talking unprecedented disaster. Most of our food, and most of our food animals, uses corn in one stage of the manufacturing process or another.
So, yeah, that's kind of the big thing about climate change. It isn't some big, apocalyptic weather event that wipes out a city... it's all these little changes. All these various different weather patterns getting more intense and more off the rails, until something fails in our infrastructure or agricultural base. It won't wipe out humanity, but a lot of people can still die or get hurt, and our overall standard of living as a species begins to drop.