There are several groups with reasonably widespread chapters trying to push climate action:

• Sunrise — youth-oriented, pushing the Green New Deal. US only. Find a local hub here. Email the hub organizer to get involved. They're volunteers, and often busy, so follow up if you don't hear back.
• Citizens Climate Lobby — broader age range, studiously bipartisan. In the US CCL is pushing a revenue-neutral carbon tax and dividend bill, H.R. 763You can find a signup form for Citizens Climate Lobby here.Make sure you figure out where the monthly meeting is and attend.
• 350.org — This is the biggest and oldest climate group. They're involved in a variety of actions, ranging from divestment to lobbying for state/province level and municipal legislation. Broad age range. Local groups can be found here
• Extinction Rebellion believes in the use of nonviolent civil disobedience, including a willingness of large number of people to be arrested, on a large scale to create political change. They are most active in the UK, but also have a significant number of active local chapters in the US and other countries. Local chapters are mostly listed here but some in the US are only listed at the bottom of this page.

If you want to find one that works for you, go down the list (and check the comments) and find out which ones are active near you. Attend a meeting or action or two to get a sense of what the group is like, and then start doing more to help.

There are others, and depending on you and your community, another group might be the best choice. If you don't feel that one of these group is a good fit for you, tell us where you are and what your community is like, and ask for help.

If you think there's something significant that one of the big groups isn't handling, ask about it. Maybe somebody can help you figure out how to get it done.