Hi Josh, welcome to the forum.
I am off Twitter at the moment so that I can focus on Google+.
Here's my approach when it comes to building a genuine following or relationships on any social network. As other members have mentioned, post quality stuffs to your stream. However, doing so will not get you far if you don't have a lot of followers. Apart from posting useful tweets, follow and interact (replies, retweets or even asking them questions) with people who are responsive on these social networks. Once they get to know you better, they will then start to follow you as well. With this, you can be sure that you are only getting genuine followers.
Of course, just like ice-breaking sessions with strangers, it is not easy to find someone that "clicks" with you. Some may take a longer time to evaluate your profile and tweets while some will follow you back right away. Be persistent, be genuinely interested in other people and you will start to see a slow and steady increase in REAL followers.
Hope the reply above is useful, Josh. Feel free to ask me questions.