They would witness to people, bearing their testimony. They were genuine, honest people that were hard workers. Missionaries being sent out preaching the gospel.
For the first 30-50 years there was much persecution, even "witch hunts"(as some could call it) on the Mormons which took the lives of many LDS people. After that, when people became aware of the constitution, the persecution remained verbal and not so much deadly.
Well, missionaries were sent out a few years after the official organization. Before that point, it was from personal testimonies being told to people.
Here is an interesting site that shows the growth of what we call stakes. They are bigger geographical areas that include 3-4 congregations together.
The LDS church continues to grow across the world. As Joseph Smith prophesied:
"...the standard of truth has been erected; No unhallowed hand can stop the work from progressing; persecutions may rage, mobs may combine, armies may assemble, calumny may defame, but the truth of God will go forth boldly, nobly, and independent, till it has penetrated every continent, visited every clime, swept every country, and sounded in every ear, till the purposes of God shall be accomplished, and the great Jehovah shall say the work is done."
In 1820, there was no church. Joseph's family always believed him. They stood by him consistently. The church was officially formed with six members on April 6, 1830. There were many more present at that time and before the day was over, I believe there were about 25 baptisms. By 1835 there were a few thousand members. By 1840 there were over 10,000 members. By 1850 there were over 30,000 members. The first million was reached in about 1950. Since then, the church has expanded to over 13 million.