Had a crippling fear of death as a child. Would literally make myself physically sick thinking about it. I would cry when we drove past cemeteries. It was bad, man. I couldn't reconcile that the lights simply go out one day and that's it. No more consciousness.
I eventually came out of it. There was no one pronged approach to it. But here's what got me started down the path of acceptance that this all ends one day:
-Death can be a release from the pain of living. Maybe you're old and have cancer. Maybe you've watched everyone you know die and you're alone. Maybe you're just tired of the daily routine once you're good and old. Death is an escape from all that. It comes blessedly to some people. Some people die happy, having lived a full life and they're just.....ready to go. Simply knowing that death can be a blessing did a lot to assuage the anguish I felt knowing that I (and fucking everyone) dies.
-I got spiritual. I want to know what "it" means, if "it" means anything at all. I want to live another life as someone new, to do a better job than I did with this one and give them what strength and wisdom I acquired in life. I like to believe that at the end of the day, everything we are just reverts back to energy in the system and that brings me a great deal of peace. Even if it turns out I'm wrong about errything, like I'm going to a Hell I didn't believe in, judged by a God I didn't worship....it's still something greater than the sum total of my living experience, and I look forward one day to crossing the threshold. It's not called "the last great adventure" for nothing.
-My grandfather, a pastor for the latter half of his life, did not go easily. I'm told he was genuinely terrified on his death bed, despite his faith and conviction that he was headed for a better place. I don't want to be that when my time comes, assuming I even have a say in the matter. I want to go peacefully, with no regrets. I want to greet death with a smile, not a grimace or a wail. And by the time I am actually dead, well, things like embarrassment and shame are already behind me.
-This is perhaps the most important thing I learned though: you can't change it. You can't control it. You can fight it, try to dream it away, or ignore it until it's right on top of you, it doesn't matter. Much like the things we stress about in life that aren't worth stressing over because we have no control over traffic, asshole bosses or coworkers, the economy and all other things....death is at the top of the list for "shit you can't do anything about so why are you letting it bug you?" The fact death is bothering you might speak to some other actual problem you have with things you can't control, and death has crossed your awareness as the biggest and most obvious thing you can't do anything about.
-Channel your understanding that your life matters, even on the most basic personal level, into something useful or enjoyable. Learn. Get skilled. Reach a pinnacle. Explore. Live. Love. The knowledge of your own mortality can drive you into a fear-based life style if you let it. But the only fear that I think is worth letting motivate you is the fear you're not doing with your oh-so-very-finite life what you want to do with it. Whether that's being good at something, being popular, being worldly, being successful, rich, educated, or yes, even just getting to slack and have as much fun as possible, it's all a better use of your energy and literally your existence than living trapped by fear and dread.
-Personally, I made a friend of death and that made me a pretty morbid teenager. Not quite goth but not exactly the life of the party. That's not always a healthy thing either. Being obsessed with death doesn't make you very likeable or relatable to people, and may scare the shit out of some people. Depending on your age, people might view it as borderline something-they-need-to-intervene-in. Don't get sucked up in the question unless you're going to do something useful with it. Just brooding about the finality of it all and wearing black because colors are for people who don't realize they're already dead and saying "nothing matters anyways" in response to most questions is NOT the way to go.
I guess my biggest tip is: don't fear death. Learn to respect it. Learn to understand its place in the entire cycle of existence, how without death nothing could live because life is death, and death is life. You don't even need to be religious OR spiritual to appreciate this truth, you understand it via science or just basic, everyday observation.
When you can grok that death is as much a part of life as life is of death, and you can accept your place in the cycle, peace will come a little easier.