There are many groups that you can become involved in. If you're a pro-market-solution type, the Citizens Climate Climate Lobby has been around ten years, and it currently has a bill in Congress that has bipartisan support (ahem, one Republican congressman): The Energy Innovation and Carbon Dividend Act (http://energyinnovationact.org/). With this plan, all the revenue from a carbon tax is directly returned to citizens as a yearly check--no enlargement of the state.
Carbon pricing (which can come in the form of a tax or cap and trade) is the single most effective mechanism to address climate change. The idea is to internalize the _real_ costs of climate change into the price we actually pay--ramping up the price on carbon over time until it is prohibitively expensive to use fossil-fuel-expensive products, and incentivizing the economy to adapt. That fundamental price signal, where renewable energy becomes cheaper relative to fossil fuels (and similarly less fossil fuel-intensive goods are cheaper relative to fossil-fuel-expensive ones) reverberates throughout the economy.
If you're skeptical of Republicans ever coming to the light, even when they have pro market solutions like the one above to choose from, the Green New Deal is the way to go. Like Obamacare, it will require a progressive majority to push through. Frankly, this is where I'm at, after spending the better part of 5 years trying to get conservatives to take action. I've seen the general conservative position shift from "its not happening" to "it's not human-caused" to "who cares? it'll be great for the economy." I am skeptical of Republicans both a) taking action at all, and b) taking action at the scale REQUIRED to address climate change (i.e., a high price on carbon that is not meddled with by oil baron lobbies).