Early Christians had a particular vision, and that vision needed to spread, so the mechanism was:
Accept Jesus, then get others to accept him.
This is similar to the vision of our form of agriculture:
Grow, then get more land, increase food production, then grow some
These two quotes came from Beyond Civilization, by Daniel Quinn.
So, we start with the fact that we want our vision of God to spread. You can see this in a similar way with other religions.
Now, do you have to buy into any one particular religious belief, so should Catholics try to convert Baptists? I think that as long as the vision is similar to Jesus' then we should focus on the two greatest commandments which is on how to properly relate to God and each other, with love. So, some people want to force their views on others that don't have their own narrow view of the world, but that seems geared more toward personal glory (bragging rights) than trying to help share Jesus' vision.
Now, you mention in a comment why try to convert people that are happy. Some people feel that they are happy as drug users, or skipping school. If I truly love someone then I would like to see them happy for eternity, and Jesus is doorway to this, then I should share my beliefs, but, as I mentioned in another of your questions, being pushy will just drive people to put up walls and become more resistant. Being pushy wasn't the way that Jesus nor the early Apostles approached this, and using them as a guideline would show that being pushy is probably for the wrong motives.
So, why not share our view on relationships with God and others, and show how it can help them, but that starts with understanding that person, not just trying to use a one-size fits all approach.
We can share our beliefs without annoying people, if we go slow, let God direct us, and understand the person we are talking with.
One of the best books to help show how this works is Dale Carnegie's How to Make Friends and Influence People, btw, if you don't want to use a religious writing.