It’s funny, Jesus got asked almost exactly this question once. Check out Mark 12:18-27. (You can find a bible online at http://www.biblegateway.com if you don’t have one handy.)
Some people who didn’t believe in an afterlife came to Jesus and proposed a scenario where a woman ends up getting married to 7 different guys because all of her husbands die. (Unluckiest woman ever!) And they ask, so who’s wife will she be in the resurrection? And Jesus tells them that in the resurrection people will “neither marry nor be given in marriage, but are like the angels in heaven.”
So, according to Jesus, people won’t be married in heaven. That doesn’t really answer your question about relationships more broadly though. But one of the things about marriage, and one of the reasons that there won’t be marriage anymore, is that it is supposed to show people what heaven will be like anyway. I know that a lot of marriages aren’t any good, but that’s not how it was intended to be. The relationship between and husband and wife are to reflect the same relationship between Christ and his church. Marriages are supposed to be SO good that they show you how close God is to us, and what it is like to be in a relationship with him. Check out Ephesians 5:25-32.
In the end, people were created to have relationships - with God and with each other. They are part of God’s plan for this world, and they will stay part of God’s plan for the next world too. Some things will change, like marriage. But in its place the Bible leads us to expect a deep, satisfying relationship with everyone.
The Bible’s picture of the resurrection is one of true happiness:
“He will wipe away every tear from their eyes, and death shall be no more, neither shall there be mourning nor crying nor pain anymore, for the former things have passed away.” (Revelation 21:4)
As well as happiness, it shows us true community:
“After this I looked, and behold, a great multitude that no one could number, from every nation, from all tribes and peoples and languages, standing before the throne and before the Lamb, clothed in white robes, with palm branches in their hands…” (Revelation 7:9)
I’m afraid that the Bible doesn’t get much more specific than that. We don’t really know the details of how we will relate in heaven, or what it will be like specifically. But we know that it will be a good community that works the way communities now are meant to work but don’t. God is fixing this broken world - including our relationships. It all starts by fixing our relationship to God, which Jesus does for us on the cross. Then once he puts that right he fixes our relationships with each other as well.
My own personal feeling about this is that we will not forget the close relationships we had here on earth. They will still be there but even better - we won’t hurt or disappoint each other any more. However all the relationships we have in heaven will be like that - close, loving and fulfilling. On a more humerous side I see heaven as the place where I will have all the time in teh world to catch up with everyone and the memory to cope with it.
Hope that helps