1. Unlike what reddit loves to say, politics are not as simple as "conservatives want to keep things frozen in time, progressives want to advance society". Both sides want to advance, and I'd say the main difference is that conservatives tend to prefer cautious bottom-up changes while the progressives love radical top-down changes when it is for what they consider a good cause.

2. It's not true that conservatives don't care about climate change. It's just that on a political level in North America the left has dominated that field for so long that the right can't even set a foot in it.

But if you look at global conservatism (in Europe for example), they do care about climate change, they just disagree with the left about how to approach it. The left usually poses that capitalism and consumerism are the problem, and propose global treaties that attack it on a global scale. Conservatives disagree that capitalism/consumerism are to blame and propose that instead of letting the problem rest on the hands of international committees, you approach it on a community level, fostering local sovereignty, holding ordinary people accountable while also empowering them to act better and protect the environment in their own neighborhood. See "Green Philosophy" by Roger Scruton. He explains that sometimes we are stuck thinking about global co2 emissions while not realizing that we are destroying our own homes.

Conservatives are also rarely fans of solar energy and the likes. They usually push for nuclear energy.