I hear a lot that God wants to have a relationship with us, but I'm not sure what they mean by that. To me, 'relationship' isn't the same as having feelings for someone, it means there's interaction and two-way communication. For example, if you send love letters to someone everyday but you never hear back from them, you don't have a relationship, you just have feelings. If you send someone love letters and they like you back, but they don't respond to your letters and meet with you for coffee and conversation, then you just have the potential for a relationship, not an actual relationship. And on the flip side, if you don't have strong feelings toward someone but you interact and spend time with them everyday, and they with you, then you have a relationship, though not a very passionate one.
I'm wary of the idea of hearing God's voice, having visions, or other spiritual experiences. Not that they don't ever happen, but I don't think we're supposed to look for signs and wonders, and I think doing so sets us up to be deceived. Given that, I think it seems more like we have the potential for a relationship, but that it will only be realized when we meet him in heaven. But I feel heartbroken to say that. Is it possible somehow to have a relationship with God in this life?
A very good question. Your concept of relationship is good, and your thoughts about hearing God's voice are good. Fortunately, you can have that kind of two-way interaction with God, except that it doesn't necessarily happen whenever you want, but when God chooses. (You can ask, but his ultimate goal for you is that you trust him even when he doesn't speak [or anything else in all creation happens]. Fervent persistence in seeking his voice may be due to lack of faith, which he may choose to not reward.)
Whoever has my commands and obeys them, he is the one who loves me. He who loves me will be loved by my Father, and I too will love him and show myself to him.” (John 14:21, 1984 NIV)
God can communicate to you in many ways. It can be by voice—if you are new to receiving supernaturally from God, my recommended rule of thumb is: if you wonder if it is from God or not, it is not. (It takes some people decades to distinguish God's voice from all the other voices.) God can, if he chooses, speak to you so the question of who said it doesn't even arise in your mind. Remember Matthew 7:18 and Luke 11:11-12. (God doesn't really say things that push you away from him.)
However, God can communicate to you in many other ways, and I think they are more common.
1. A thought comes to mind that shouldn't be there. My experience is that it doesn't feel like someone putting a thought into my mind, but like a thought I had on my own. We are joined with Christ; he is not external to us. Sometimes you can recognize that there is no reason whatsoever a thought should have popped into your mind, such as when you didn't have the information required to reason through to that thought.
2. A big one is that your prayers are answered. Be sensitive to exactly what you are asking God for. If you are asking him to do something a certain way, you may not
see an answer to it. Examine why you are asking for something, then ask God to satisfy that need somehow; leave the "how" up to him (as well as the time frame, but keep asking).
3. Possibly the most common experience of God is that your understanding of a passage of Scripture clicks in a deep way. This tend to happen after we have studied, pondered the passage a while, and have experienced what the passage speaks about. We understand everything/anything through our experiences. Understanding Scripture comes when we connect what we're reading to something we have experienced. (Being obedient to God gives us the experiences that give us understanding of Scripture and helps us know God himself better.) Less common is when our understanding of a passage suddenly appears for no reason. You might want to hold such new understandings at bay for a while. Satan inserts an incorrect understanding of something into our minds a lot. It is good to discuss such new understandings with other Christians.
4. Other ways. God can communicate with us any way he wants to. You can hear thunder in the clouds and you can recognize that God spoke to you because you understood what the thunder meant. He can speak to you through other people, even unbelievers.
In all cases, reject everything you think is from God if it is contrary to Scripture or Truth. (One reason for ongoing study of Scripture, because God won't speak to you if it is going to harm you.)
Having said all that, consider why you want to hear God supernaturally. Most of my life, I've heard Satan hundreds of times as much as hearing from God. Hearing from God is special—it is a miracle—and it would be bad for us to start considering it commonplace, which leads to considering God commonplace, and that is very, very bad for us, for he is holy. Our goal must be to know God through Scripture. Remember that where God's voice and Scripture conflict, Scripture always wins. God may test you in this.
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God, because many false prophets have gone out into the world. (1 John 4:1, 1984 NIV)
There are many, many saved people who do not ever have supernatural experiences of God. The most common reasons IMO are: (1) a person doesn't really want them, or it would be bad for God to give them to you. Your faith in the truths/God of Scripture, and submission to him, must be firm, otherwise an experience of God by "sight" may harm you terribly (Satan's sins are not forgivable because IMO he rejected god by sight), and (2) they haven't accepted the word "all"
in their hearts seriously enough in Mark 12:30-31. (e.g., Is God in your thoughts throughout the day?)