The case for Christianity consists of different layers. Believing the gospel requires believing miracles can exist. Believing in miracles requires believing in God. The existence of God is the foundation level of the building. How do we get people to believe in God?
Different people will respond to different arguments for the existence of God — that's why there are so many of them. You should have a familiarity with as many as possible, but you may not be able to argue some as convincingly as others. If someone seems inclined toward an argument you're weak on, you can always point them to a book or article, but to be credible, you need to be able to make a case for your own belief.
This is the case I would make:
There are four distinct arguments for the existence of a god I find compelling. While they may resemble each other, they are independent of each other. (Since entire books have been written on this topic, I will offer an abbreviated version of the arguments and point you to more in-depth treatments if you're interested in going further.)
1. The cosmological argument
Every effect must have a cause. The universe began, that is, it is an effect. Therefore it must have a cause. That thing that caused the universe is what we will call God.
Why should we believe the universe began? Physics. Among the reasons are the second law of thermodynamics, general relativity, and Hubble's evidence of an expanding universe. In brief, this universe looks like the remains of an explosion. That explosion occurred at a specific point in time before which our universe did not exist.
Something has to be eternal. The universe isn't it. Whatever ultimately caused our universe is.
Yes, there are cosmologists driving themselves nuts trying to prove the universe didn't get caused or was caused by some natural phenomenon. There are two things to point out about that: 1) They are completely untestable (aka, "not science"). 2) The researchers in question largely admit freely that they are pursuing these things for the sole purpose of getting rid of the moment of creation and therefore the creator. It is not science but philosophy that drives this.
(There are tons of books that cover this. A short, cheap one is the Pocket Handbook of Christian Apologetics.)
2. The design argument
Our universe, our planet, and life itself show signs of being made very precisely so that life (in fact, intelligent life) can exist. Someone has made all of this possible.
As I posted earlier, before I had ever heard of Intelligent Design, I had been introduced to the concepts via physics professors. Once I came across the totality of the evidence it was pretty stunning. There are dozens of features of the universe that do not have to be any particular value, but for life to exist — any life — they must fall within very tight ranges. Among them: the energy of the big bang, the speed of light, and the ratio of the electromagnetic force to gravity. The last time I checked there were 41 physical constants on the list. Roger Penrose put the odds of the big bang producing a universe fit for life at 1 in 10 to the 10 to the 123 (1 in 10^(10^123)). It is impossible, and yet here we are.
The earth itself shows the same kind of design. Lately the news has been full of the discovery of "earth-like" planets, but you have to understand how loosely they use the term. For life to exist you need a planet in the right part of the right kind of galaxy, orbiting the right distance from the right kind of star, and with a satellite of a certain size. Not only are certain chemicals needed (or needed to be absent), those things require the planet to have a certain size and certain physical characteristics. Life is far easier to kill than to sustain, so it takes a very special planet to provide a place where any life — much less intelligent life — can thrive. There really is no reason there had to be even one.
Finally, the very existence of life is a puzzle. Listening to many scientists, you would think life is so easy to build it should be around every corner. It's not. For natural processes to create life, you need a very special group of chemicals kept in a very careful balance. You need these chemicals to create a self-replicating system than allows errors to creep in. You need some of those errors to not be toxic, though most are. You need something to protect that life from the very forces that created it. The truth is the universe is not old enough for natural process alone to have produced microbial life, much less us. If there is no God, our existence is more of a miracle than if he exists.
It was the design argument that convinced Anthony Flew of the existence of God. The cumulative case for design to get us here is so great that the universe has been described as a "put up job" by some scientists.
(Here you really need more depth, so I recommend dedicated books like The Creator and the Cosmos or some of Ross' other works. Destiny or Chance, though not Christian, is a good book on the topic, too, if you don't mind his naturalistic conclusions.)
3. The moral argument
We know that some things are morally wrong. How do we know that?
I don't mean that some things are a terrible way to run a society. I mean some things should never happen. We have no problem saying the Holocaust was wrong. We all see hurting children as abhorrent and condemn people like the Indian gang rapists and Adolph Hitler with every bit of righteous indignation we can muster. There are people who don't seem to possess a properly developed moral sense, but, like color-blindness, the defect only serves to highlight the norm.
We all believe that there is an objective standard for right and wrong. Even those who claim they don't can't live like that.
Where would such a thing come from? Valiant attempts at an evolutionary explanation have been made, but they all fall short because the thing we know we should do is often, from a survival of the fittest (or survival of our genes) point of view, the last thing we should do. The simple truth is we all possess knowledge of a moral standard that comes from outside of us. The only explanation is that it is imparted to us by our creator.
CS Lewis made this argument famous in his Mere Christianity.
4. The argument from religious experience
The argument here is not that people's religious experiences are true. It is simply that everyone has them.
This is another argument handled ably by Lewis (again, Mere Christianity), so I will simply borrow from him:
Hunger presupposes the existence of food. Thirst presupposes water. Humans yearn to fulfill a spiritual need that we really can't explain, but its presence points to the existence of Something that can fill it. As Lewis put it, "If I find in myself a desire which no experience in this world can satisfy, the most probable explanation is that I was made for another world."
The universality of religion points to the existence of God. It doesn't matter that different peoples have gone about it different ways. It is that every people has gone about it. In what other situation do humans feel a deep need for something that doesn't exist?
Of course, all of these arguments are for A god, but not necessarily for the God. Rather than argue directly for God, I would argue for Christianity, with the Christian conception of God included. What follows depends on one of the previous arguments being successful. For someone to take the Bible or Christianity seriously, they have to be open to the possibility of the miraculous. This, then, is the second layer of the argument, depending on the first:
5. The reliability of scripture
If someone automatically discounts the miraculous, the phrase "reliability of scripture" is going to be laughable. But if they are open to the possibility, we can show that where the Bible can be tested, it has been shown to be historically reliable. You can probably get away with just the New Testament, but the Old Testament can be shown to be reliable, too. That doesn't mean there aren't still question marks hanging around, but it has been shown to be correct many times and it has never been shown to be conclusively wrong.
The primary goal is to show that the authors of the New Testament 1) were trying to and 2) were successful at giving an honest account of what they experienced. If they presented what happened as they saw it, we can then talk about what they thought they saw.
(Besides the books linked in the paragraph, The Case for Christ is a good introduction to this topic and the next.)
6. The resurrection
If the gospels are reliable, we can address what they say about Jesus and his death and resurrection. As I have argued before, the gospel is a story no one would make up. But even if they were trying to be honest, they might be mistaken. The question is, what is the best explanation for the resurrection and the birth of the New Testament church — a miraculous or natural explanation?
If someone doesn't believe miracles are at least possible, no naturalistic explanation is more ridiculous than an actual resurrection. But if they are open to the possibility, we can show that a supernatural resurrection is the simplest, sanest explanation.
Hundreds of people saw Jesus die. Hundreds of people saw Jesus alive afterwards. Even some who didn't believe in Jesus before hand saw him. Because of this they overturned their lives — giving up the way they were raised to follow after new teachings (in a culture that distrusted the new), preaching the triumph of an executed criminal, and risking their relationships and even lives. Every natural explanation looks ridiculous in the face of what they did, what they risked. The only reasonable explanation is that they saw what they believed to be the resurrected Jesus — in different times and places and under differing conditions, making their being mistaken ridiculously unlikely. The most reasonable conclusion is that Jesus really did get up and bodily leave his grave.
7. On that basis, we can build the third layer: If the resurrection happened, it's a short trip from there to Christianity. Jesus himself pointed to his resurrection as the vindication of all he taught and claimed about himself. It's hard to disagree — getting raised from the dead is a pretty good sign God is on your side. Even if you have trouble believing in some biblical passage or facet of Christian teaching, the major pieces are pretty well in place, and that includes the Christian conception of God.
There is no other religion with scriptures of the same character and quality as Christianity. There is no other religion with a huge, public, defensible miraculous foundation. Christianity and it's conception of God win any contest.