Why is believing the correct thing important?

It depends on who you believe God is. The Christian God is love, grace, mercy and righteousness. He has allowed many forms of knowing Him, they are all faulty and true - each to varying degrees of each. There is no perfect truth in this age and all will die with much error. He has not designed such a wide variety of groups, all with error - this is the result of a fallen/corrupted world.

But we would believe (to varying degrees) that there is a plan to allow all of humanity to know the true God and choose the way He has provided to enter the next life - to be with Him and be free of sin and death.

Salvation in Christianity is not about what we know or how much truth we have, it is about how we live and how willing we are to abandon our earthly treasures for the heavenly ones. We all get opportunity to choose on a daily basis. Will we be getters or givers, will we be selfish or selfless, will we be patient or not, generous or not...

When we've finished with this life, we will not be perfect or complete or able to go to the next - except for the way that Jesus has offered us all. In his righteousness alone, he has defeated sin and death for all.

Every religion has a God. Some are harsh and unforgiving, some are demanding and capricious. Our God requires much of us. But what He requires, He provides. This is unique to Christianity and is at least partially recognised in all branches.

He requires love, sacrifice, righteousness, holiness, belief, faith, trust.

He has provided these things in Christ. And He calls each to follow as best they can, knowing that all stand by grace alone and not by merit in any good we have done or truth we think we may know.

It doesn't matter if you believe the wrong stuff. God can straighten you out in a moment should He choose to do so. Until now He hasn't, so it can't be that important for this life.

God is bigger than Mormons and Baptists and Catholics and... frankly I don't think He cares much about any of them or we wouldn't be in this mess. But there IS a plan to fix everything and it isn't finished yet. In fact, it's hardly started!

So Jesus was saying to those Jews who had believed Him, “If you continue in My word, then you are truly My disciples; 32and you will know the truth, and the truth will set you free.” John 8:31

The truth does matter, but it is not critical for salvation. Freedom in this life is relative to the truth we encounter - if we can recognise the truth when we see it.

They are not of the world, just as I am not of the world. Sanctify them in the truth; Your word is truth. Just as You sent Me into the world, I also sent them into the world. And for their sakes I sanctify Myself, so that they themselves also may be sanctified in truth. John 17:16

Edit regarding truth not being critical,

The truth about salvation through Christ is of utmost importance.

What is not critical is whether one believes in a trinity, a mode, an angel or whatever. This confusion is not of God and is part of what Jesus came to fix, the job is not finished yet as the deceiver is still deceiving.

Jesus made the stakes clear:

31 To the Jews who had believed him, Jesus said, “If you hold to my teaching, you are really my disciples. 32 Then you will know the truth, and the truth will set you free.” (John 8:31-33)

The stakes are freedom.

The Bible teaches a process of sanctification that proceeds from a transformation of the mind (learning the truth, the domain of the Son of God), to a reformation of the affections and emotions (the domain of the Holy Spirit), to an alteration in priorities and habits of action (the domain of the Father), leading to a harvest of righteousness (good deeds) and ultimately a life of peace. You see aspects of this process described in the Parables of the Four soils and the Wheat and the Tares, and many other places. The line is drawn: lies of the devil (the Tares or weeds) grow up fast and look impressive but yield no good fruit, no grain for the harvest. Truths from God yield good fruit. The truth is the seed - no truth, no good growth.

The stakes are fruitfulness.

Another area is loyalty. Jesus is loyal to the people he "knows", as revealed in Matthew 7. It is about relationship. Truth is not only propositional, it is personal. Different religions may share some of the same propositions, but not all share the same person.

Jesus answered, “I am the way and the truth and the life. No one comes to the Father except through me. (John 14:6)

Jesus is not just one who knows the truth and teaches the truth, he literally IS the truth. He knows his sheep and calls them by name, and when called by name, his sheep know he is their shepherd.

10 “Very truly I tell you Pharisees, anyone who does not enter the sheep pen by the gate, but climbs in by some other way, is a thief and a robber. 2 The one who enters by the gate is the shepherd of the sheep. 3 The gatekeeper opens the gate for him, and the sheep listen to his voice. He calls his own sheep by name and leads them out. 4 When he has brought out all his own, he goes on ahead of them, and his sheep follow him because they know his voice. 5 But they will never follow a stranger; in fact, they will run away from him because they do not recognize a stranger’s voice.” 6 Jesus used this figure of speech, but the Pharisees did not understand what he was telling them.

7 Therefore Jesus said again, “Very truly I tell you, I am the gate for the sheep. 8 All who have come before me are thieves and robbers, but the sheep have not listened to them. 9 I am the gate; whoever enters through me will be saved. They will come in and go out, and find pasture. 10 The thief comes only to steal and kill and destroy; I have come that they may have life, and have it to the full.

11 “I am the good shepherd. The good shepherd lays down his life for the sheep. 12 The hired hand is not the shepherd and does not own the sheep. So when he sees the wolf coming, he abandons the sheep and runs away. Then the wolf attacks the flock and scatters it. 13 The man runs away because he is a hired hand and cares nothing for the sheep.

14 “I am the good shepherd; I know my sheep and my sheep know me— 15 just as the Father knows me and I know the Father—and I lay down my life for the sheep. 16 I have other sheep that are not of this sheep pen. I must bring them also. They too will listen to my voice, and there shall be one flock and one shepherd. 17 The reason my Father loves me is that I lay down my life—only to take it up again. 18 No one takes it from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again. This command I received from my Father.” (John 10:1-18)

The stakes are friendship.