Thank you, first of all, for asking. This is the first thing you have done to be able to successfully minister to me. I appreciate that you are interested in hearing how I would wish to be introduced to your religion, instead of being introduced to it on your terms. I cannot understate how important this is, and how much I appreciate that you think this way.
Second, I want to mention that if we live very long in the world, among other people and in hearing range of any media, we are already aware of Christianity. There might be someone somewhere in the deep dark forest who still wears leaves for clothing that somehow hasn’t heard of Christianity, but that is not anyone you are likely to find on a regular basis and consider as a candidate for “sharing the good news.” In fact, you may be talking to someone who has already been made very familiar with Christianity, and found it to be not what they needed, and they have moved on. It is important to know that if you choose to deliver your message. Miracle stories from the Bible, or from your life, may be things that we have heard but do not believe in, and have in fact been told before. I, as a non-believer, would want to know what (if anything) you have to add to this background that is new, different, and potentially true.
People who have already heard and rejected Christianity also probably have a good reason. Maybe it’s a lack of evidence, for example, or the fact that his or her brain just works differently. If you are talking to someone of a scientific bent, you may wish to de-emphasize a Biblical literalist view, because they will not accept it.
You say, “I believe it’s important to share what I believe to be the best thing to happen to me in my life, because I don’t want to withhold something that was so good for me, from other people who might need it.” As you’ve seen above, all of us (pretty much by definition) also have a “best thing” to share. Are you open to hearing about it? It could be a book, a spiritual practice, a story about meeting our sweetie, or anything. Will you listen? Will you find common ground? Nobody likes to be preached down to, and no one likes to be the object of condescention. If you approach on a level that says you are open to really hearing, I might be more willing to listen to you in return.
And this. If you are able-bodied,affluent enough to care for your own needs (and you have the Internet, so I am guessing you also have things like food and clean water), you have been given some very awesome things. I would be interested in knowing if you share your other awesome things with people in need, too. In other words, do you walk your talk? Is Jesus the only thing you share, or do you do also help the people who have less? To people in desperate places, a loaf of bread or gallon of clean water is more wonderful than hearing about Jesus, and as you do for the least of them, you also do for Him. (I have a very good Christian friend, maybe the best Christian I know, who has devoted his life to bringing water purifiers to Africa. If I am going to listen to a Christian, I will listen to one like him.) If all I see you offering to others is the “Good News,” I may possibly judge you as having more of a need for self validation (or will to power) than true selfless intent. Walking the talk will make your message seem far more real than hot air.
And lastly, remember the parable of the sower. Some of your grain will fall on fertile ground, and those people will be really energized by your message in a way that no Christian has ever energized them, and they may take up your words and grow. Some people will be like the rocky soil, and you can scatter as much seed as you want there but it will not take root. if you can ascertain which listeners are like fertile soil and which are like rocky soil, you will be able to spend your resources where they will have the most result for you. Yelling at rocky soil (or someone who is truly uninterested in your good news) will not make the seeds grow – and may even do more harm for your message than good, because the next time a Christian brings up the same message (maybe even the one who would win them over!), they will remember you as one more blowhard that gave Christianity a bad name. On the other hand, if you thank us politely and continue on, we will remember that there are nice Christians, and maybe the next time we hear the word from someone else who wants to share, we will think favorably upon it.
Again, thank you for asking! Would that more of your brethren were listeners than talkers.