This advice works for me 100% of the time, but may not work for you. Whether it does or does not I'm going to present it as if it's the Golden Rule of life and hope it resonates with you.
1. The key is to pay attention to your own thoughts. Simply notice what you're thinking about. You can't be in this mode all the time, but when you do catch yourself thinking, observe the thought you just interrupted.
2. Don't judge your own thoughts. Don't even judge your own cynicism. This non-judgmental attitude is the first step to changing anything about your behavior and your state of mind, even if that seems counter-intuitive. Almost everyone has a resistance to that which assumes authority to judge. Don't assume you are the authority of your mind. There are so many reasons why you aren't, but the biggest reason is that you have no control over your subconscious mind. "You" lives in the conscious mind, and "You" believes it has control. But your brain sends a signal to your arm to move it half a second before "You" can even think of claiming responsibility for the action. Your mind is like a fluid. Stuff rises from the depths and sinks back down again and you mostly don't control that. It's a natural process. So there's no reason to judge anyways.
3. Ask yourself lots of questions about your thoughts. "Why" is the most important question. Explore your thoughts without judgement. Try and find the answers to these questions. You're allowed to speculate and you're allowed to get it wrong. It doesn't really matter.
4. Choose the thoughts that present consciously. This is the aspect of your mind that "You" has control over. If you ask enough questions you'll eventually be able to weed out the thoughts that don't do you any good. Simply choose not to think about them anymore. Don't try to get rid of them (that only draws more focus to the thought you don't want to think, instead simply tune in to a different station. If you find a cynical thought come to mind, just think about it a bit more deeply. Even if the thought is "true", you could look at your subject from different angles and find a less cynical perspective, one that is equally true, and find something about it that gives you energy or reminds you of what makes you happy.
5. Some thoughts come from a damaged place. If you find a wound, you'll have to address it. Once again, don't judge the thought that comes from the wound. You'll need to accept it and give it proper attention if you want to heal.
6. Sometimes certain thoughts just need to be ignored. Sometimes thoughts are meaningless and toxic. Just ignore them. No need to think about it more deeply if your brain is muttering about how much of a bitch that lady at the grocery store was. It's obviously nonsense. Learn to recognize thoughts that deserve your analysis and those that don't.
Cynicism is a delicate thing and often results from fear. It can be healthy in environments that encourage thoughtlessness, but toxic in environments that require hope, faith, relaxation and creativity. The biggest thing you should keep in mind is that everything can be seen from multiple different angles. This is why it's possible for a thousand people to hold a thousand different opinions on the same issue-- stuff is multi-faceted. Sometimes, there's no objective truth. You can choose to look at anything from any angle you want and find other truths even in the same object, thus immediately changing your state of mind in that direction.