This is a very broad question that can be approached from various perspectives, so I probably can't do it justice. But let me point you in the right direction and provide some further resources for you.
1. There are philosophical arguments for the existence of God such as:
a. The ontological (highest ideal) argument
b. The teleological (intelligent design) argument
c. The cosmological (first cause) argument
d. The Kalam (time) argument
e. The Axiological (moral) argument
I cannot explain each one in detail, but read this post carefully and/or grab a copy of Mark Driscoll's On Who is God? Both sources deal with these arguments in detail.
2. God reveals Himself to us in two ways:
a. General Revelation: God reveals himself to everyone everywhere through general revelation, which includes*:
i. Creation: Through all of creation (galaxies, atoms, earth, sky, animals, flower, etc.), God reveals his creativity ability, power and goodness to us (and thus his existence). This is why Romans 1:19-20 tells us that we are without excuse in knowing that God does exist.
ii. Common Grace: This is a term St. Augustine used. "Common Grace" refers to God's general goodness to believers and unbelievers alike. God's common grace allows all people to learn and develop in the areas of science, philosophy, technology and medicine. It also allows for families to exist and for societies to thrive. Lastly, common grace allows people who are not connected to Jesus Christ to live decent moral lives of compassion and kindness.
iii. The Human Conscience: God reveals himself through the conscience. We all know it is wrong to murder, steal, cheat. Why? Because God has written morality into our very being (Romans 2:14-15).
b. Special Revelation: In addition to General Revelation, God shows us his existence to us through special, supernatural means. Listen to what GotQuestions says:
Special revelation is how God has chosen to reveal Himself through miraculous means. Special revelation includes physical appearances of God, dreams, visions, the written Word of God, and most importantly—Jesus Christ. The Bible records God appearing in physical form many times (Genesis 3:8, 18:1; Exodus 3:1-4, 34:5-7), and the Bible records God speaking to people through dreams (Genesis 28:12, 37:5; 1 Kings 3:5; Daniel 2) and visions (Genesis 15:1; Ezekiel 8:3-4; Daniel 7; 2 Corinthians 12:1-7).
Of primary importance in the revealing of God is His Word, the Bible, which is also a form of special revelation. God miraculously guided the authors of Scripture to correctly record His message to mankind, while still using their own styles and personalities. The Word of God is living and active (Hebrews 4:12). The Word of God is inspired, profitable, and sufficient (2 Timothy 3:16-17). God determined to have the truth regarding Him recorded in written form because He knew the inaccuracy and unreliability of oral tradition. He also understood that the dreams and visions of man can be misinterpreted. God decided to reveal everything that humanity needs to know about Him, what He expects, and what He has done for us in the Bible.
The ultimate form of special revelation is the Person of Jesus Christ. God became a human being (John 1:1, 14). Hebrews 1:1-3 summarizes it best, “In the past God spoke to our forefathers through the prophets at many times and in various ways, but in these last days he has spoken to us by his Son … The Son is the radiance of God's glory and the exact representation of his being.” God became a human being, in the Person of Jesus Christ, to identify with us, to set an example for us, to teach us, to reveal Himself to us, and, most importantly, to provide salvation for us by humbling Himself in death on the cross (Philippians 2:6-8). Jesus Christ is the ultimate “special revelation” from God.
So friend, I hope these resources help. Read them carefully. Practically, I would suggest you discern what kind of person your friend is. Some unbelievers bash Christianity (and religion in general) because they do not understand it, or perhaps they have meet some people that have been poor representatives of Jesus. These individuals have not carefully considered their arguments, so they simply say that Christianity is stupid and shallow. If your friend fall into this category, a softer, more gentle approach is required. They may need a friend who listens to them and shows them genuine care more than someone who can debate them using logical arguments.
On the other hand, there are unbelievers out there that are intellectuals/thinkers. They want to talk to someone without getting the "easy answers." For these individuals, you must be prepared. You must do good research, articulate your points effectively and be able to engage them in rigorous debate. Discernment is required here. If they are willing to listen, then continue to conversation. If they continually put you down (and you can tell they are not really trying to learn, but just want to humiliate you), then perhaps it's not time to talk to them yet.
In any case, The Bible says that we must accept by faith the fact that God exists: “And without faith it is impossible to please God, because anyone who comes to Him must believe that He exists and that He rewards those who earnestly seek Him” (Hebrews 11:6). So pray for your friends; ask the Lord to open their hearts and minds so they can have faith.
Hope this helps. I will say a prayer for you this week.
[Answered by Pastor HM]
*See Mark Driscoll & Gerry Breshears, Doctrine: What Christians Should Believe (Wheaton: Crossway Books, 2010), 37ff.
