I know lots of people who don’t believe in Jesus, and they live good lives. Shouldn’t they go to heaven?

Living a good life will not get anyone into Heaven. Jesus said that the way to enter Heaven is to be perfect (Mat 5:20, 48), not good. No one can do that. If it weren’t for Jesus, we would be hopeless. But Jesus came to save us. He is God and He came to earth and lived a perfect and sinless human life. He then took our sins upon Himself and died. In doing so, He satisfied/paid the penalty for our sin and opened the way to Heaven. The ONLY way anyone gets into Heaven is to trust in Jesus, and ask His forgiveness of their sins. He is faithful to forgive a repentant sinner, wash them clean, and give them His righteousness. Then, when they stand before God, they are seen as perfect.