There are two ways you can go to grow your follower or fan numbers on social media sites. One is paid advertising using their paid ad feature and the other is naturally growing your business by making relation with the people in your audience.
Let's explore the paid ad feature first. Do you have a Facebook page? Did you know you can advertise to people of your post/zip code area using the ad platform of Facebook. Of course this feature will help the local business more than board topic page. Whatever your topic or business is don't forget to fill the page with some good posts before calling people to visit you.
How much does it cost? It depends on your product or service. You can definitely make a return of investment and even make great profit comparing to other advertising methods. The best thing is you have make a permanent relation with the audience. You only pay Facebook for clicks or impressions. But the people who like your page will see your future posts and like again. You may also create useful posts and boost them using Facebook ad.
The other part comes growing your followers naturally and that can be done by creating good contents and getting engaged with the people. So if you own a Facebook page you can create contents and post them in related groups in Facebook. Talk to people there and let them know what you do. This of course takes longer time. But also very effective as you make natural relation with the people of your audience. Why just Facebook? You can search online and find all social media groups and pages and introduce what you offer. There are many blogs who uses Facebook comment pluging and you can make comment as your page name. So you get a backlink and targeted traffic in the same time.
It's about going to as many related places possible and the web has no limit. So you find as many popular places possible that are talking about what you offer. But the best results in the shortest period of time can be done with a mixture of both paid and natural advertising.