There are a large number of such groups and the list here hopefully will give the main ones - and be added to over time.
World Resources Institute (WRI)
Link: https://www.wri.org/
Has pages dealing with Climate, Energy, Food, Forests, Water, Cities and Ocean
United Nations Climate Change (UNFCCC)
Link: https://newsroom.unfccc.int/
United Nation - Climate Action
Link: https://www.un.org/en/climatechange
Objective (from website):
As the world confronts the pandemic and embarks on recovery, there is growing recognition that the recovery must be a pathway to a green and sustainable economy that produces jobs and prosperity, reduces emissions, and builds resilience. From the science to the solutions, this website showcases how the United Nations is bringing together all nations and peoples, so we leave no-one behind as we tackle the global climate emergency.
Climate Council
Link: https://www.climatecouncil.org.au/
Description from website: The Climate Council is Australia’s leading climate change communications organisation. We provide authoritative, expert advice to the Australian public on climate change and solutions based on the most up-to-date science available.
We’re made up of some of the country’s leading climate scientists, health, renewable energy and policy experts, as well as a team of staff, and a huge community of volunteers and supporters who power our work. As an independent voice on climate change, we get climate stories into the media, produce hard-hitting reports, call out misinformation as we see it and promote climate solutions such as the transition to renewables.
The Climate Council was founded in 2013 by tens of thousands of Australians to create a new, an independent and 100% community-funded organisation in response to the abolition of the Australian Climate Commission.
Climate Watch
Link: https://www.climatewatchdata.org/about/description
Description on their website: Climate Watch is an online platform designed to empower policymakers, researchers, media and other stakeholders with the open climate data, visualizations and resources they need to gather insights on national and global progress on climate change.
Climate Watch brings together dozens of datasets for the first time to let users analyze and compare the Nationally Determined Contributions (NDCs) under the Paris Agreement, access historical emissions data, discover how countries can leverage their climate goals to achieve their sustainable development objectives, and use models to map new pathways to a lower carbon, prosperous future.
MyClimate.org
Link: https://www.myclimate.org/information/about-myclimate/
myclimate is a partner for effective climate protection, globally and locally. Together with industry partners and private individuals, myclimate wants to shape the future of the world through advisory services and educational programmes, as well as its own projects. It does so in a market-oriented and customer-focused way as a non-profit organisation.
Mission 2020
Link: https://mission2020.global/
We’re on a mission to drive urgent action to limit the effects of climate change, particularly for the most vulnerable people and countries. With radical collaboration and stubborn optimism we will bend the curve of global GHG emissions by 2020, enabling humanity to flourish. It’s necessary, it’s desirable and it’s achievable.
Climate Action Network
Link: http://www.climatenetwork.org/
The Climate Action Network (CAN) is a worldwide network of over 1300 Non-Governmental Organizations (NGOs) in more than 130 countries, working to promote government and individual action to limit human-induced climate change to ecologically sustainable levels.
CAN members work to achieve this goal through information exchange and the coordinated development of NGO strategy on international, regional, and national climate issues. CAN has regional network hubs that coordinate these efforts around the world.
The Australian Branch 
can be found at this link: https://www.cana.net.au/
Climate for Change
Link: https://www.climateforchange.org.au/
Climate for Change’s mission is to create the social climate in Australia for effective action on climate change.
We know that social change happens when ordinary people start having conversations with those they know and trust. We support people to have effective conversations about climate change, and to take actions that will inspire others.
We are the only organisation in Australia specifically focussed on helping people to have conversations with their peers on climate change – something that is now being recognised by experts as key to building public support for the action we need on climate change.