I deal in data. I have always had an interest in history, and I got kicked out of a lot of classes when I mentioned embarrassing massacres. I learned, very young, some ways of handling harsh ideas without injuring myself. Just think of this like you do the hotpads in your kitchen: use them properly and you won't get burned.
* Don't make yourself another casualty. You cannot help anyone else if you let yourself get wrecked. Practice good self-care. Let other people see you doing this, because modeling influences behavior more than anything else.
* Choose your sources wisely. Some are more reliable than others, and some are so batshit they could put anyone into a bad mood with no gain from it. You'll need to triangulate and cross-check some stories to tell whether they're true. To minimize this workload, aim for sources that fact-check their own content.
* Be prepared to drop sources if they're just too negative. I quit if all I have to say about the content is "Da, eto Pravda" (literally, "Yes, that's the Truth," but Pravda was a Russian propaganda rag, so colloquially it means "bullshit") or "Panem et circenses" (for "bread and circuses," a means of distraction in the decline of the Roman empire).
* Keep your exposure to reasonable amounts. The more stressful material you consume, the higher the risk it will make you feel bad.
* Leaven bitter material with something sweet. I like to post photographs and recipes. I also talk about gardening, birdfeeding, and other nature stuff. Searching for those is fun. Pick whatever appeals to you. It will help other people, too, by contributing positive things to the newsflow.
* Suit the proportions to your need. When you feel more stressed, consume less of the difficult news and more upbeat stuff.
* Take breaks. Get up and stretch. Have a snack and/or a drink. Look out the windows. Connect with other people. Remind yourself that your immediate vicinity is reasonably safe.
* Look for actionable information. The world is full of problems. There is not much point worrying about ones you can't influence at all. Identify problems which can be fixed or at least buffered, then search for ways you can help. Do what you can, and spread the word about ways other people can help too.
* Actively seek good news. Some people choose to fill their newsfeed primarily or wholly with positive content. Others just try to maintain a sampling of it. Set whatever proportions work for you. Be prepared to change them if necessary.
* If you're already in a bad mood, don't read bleak news, and don't go through your friendslist bitching at people in comments. Instead, do something to lift your mood. Read neutral science news, look at LOLCATS, listen to music, read fluffy fanfic, play a game, etc. Have a list of comfort reading or other activities that make you feel better, so you can use them as needed.
* Choose some specialties. No one person can fix everything. Consume less news about things outside your areas of interest and expertise. Just skim those to keep a general awareness of the world. Dig deeper into your specialties, where you can amass more knowledge to make better contributions.
* Don't bite your tongue too much. Sometimes speaking inconvenient truths hurts less than remaining silent about atrocities or stupidities.
* When you talk about negative things, link to ways that people can fight them. Don't just dump a dead skunk on people's virtual porch without giving them a shovel to deal with it.
* Link to positive news and useful lessons. The more attention this stuff gets, the more people will do it, because eyeballs are valuable. Also, the more skills people have, the better they can handle challenges.
* Create constructive content. Teach coping skills, peacemaking, and other social engineering for a better world. Do crafts or gardening and share pictures of them. Make music or dance videos. Write stories about characters solving problems in positive ways.
* Remember that an oracle's job is to warn, not to convince. Lay out the information as best you can, and then it's other people's choice what they do with it.
* Understand that you are only responsible for what you control. If bad things are happening far away and you have no influence, they are not your fault and you shouldn't blame yourself. Same if your country's leaders have their head up a camel's ass; you don't control them, so they don't reflect on you ethically. Ignore anyone who tells you it's your own personal fault. However, if you support policies which cause certain problems, then you do have ethical responsibility in proportion to your influence. If the results make you feel bad, consider changing your stance on the issue.
* Remember the big picture. Humanity may have the ecological footprint of a comet strike, but Earth has survived those before. The biosphere might be less interesting for a while, and humanity might dwindle, but we're very unlikely to kill off everything. However, the sun will go nova several billion years from now, and eventually all the atoms will get recycled into other suns and planets. Fix what you can from where you are, but trust that the Universe will go on regardless.