I started doing bounties and growing my twitter account earlier this year. I've been experimenting and researching different ways to gain twitter followers consistently and I thought it would be helpful to others on here who might be struggling with the same issues. I also have some methods that I haven't personally tried yet as what I'm already doing is satisfactory enough, I'll go into detail on each method here and how effective I feel they are, as well as any other pros and cons that come to mind.
Obviously you'll need a twitter account, a fresh one will work but you'll want to get verified right away.
Additionally, you'll want to get an extension for your browser to follow people. This is optional, but I feel it helps organize things a lot better. I use Mass follow for Twitter, and I only use it on chrome. I don't use chrome for anything else so I only ever access my follower extension while using Chrome.
First things first, you want to configure mass follower for twitter. The main thing I use this for is to keep track of people who follow but then unfollow you later on for some reason. You will need to follow lots of people over time, but these people clog up your limit and need to be removed. I'll post my configuration for mass follow for twitter and explain what I feel is best, but it's up to you. There are some important settings though. 
To access the configuration menu, open up twitter and you will see the above image. Simply click on settings and you will get the full settings menu.
There are separate settings for following and unfollowing.
The settings for Wait and Limit are pretty much the same for each. You can wait a little or as long as you like between each follow, but I will say that doing a lot of things very quickly tends to draw attention to your twitter account and you want to avoid that whenever possible. As you can see, mine is set to around 3 seconds usually.
Limit is how many it will follow in one session. You can set this to whatever you want, but twitter wont allow you to follow more than 1000 people in a single day, and again you probably want to avoid that behavior whenever possible.
Days followed I think 3 days is about enough, but it's not too important. This is only really useful if you're mass following people all the time in the hopes of getting mutual followers, once 3 days pass the extension will remove them the next time you run unfollow.
Skip followed is important because it will ignore people you've already followed once so you don't waste your time following a known unfollower again.
Skip follower is important because you don't want to unfollow people who are following you because they may drop you as well and then you both lose. I always try to keep following those who follow me.
Mass Followed Required is the most important setting to leave checked, otherwise when you unfollow you will also unfollow bounty accounts. You don't want that to happen as it may disqualify you from your rewards.
On to the follower gaining methods:
Method 1: Mass following people in hopes of gaining mutual followers.I've tried this method, it's effective to a point but far too problematic in my opinion. Following/unfollowing a few hundred people every day definitely puts you on twitter's radar and your account can be rate limited, shadowbanned, or other annoying things. If you are still interested, just use caution.
Essentially all that is required of this method is to find active twitter accounts and follow the first 50-100 people on their follower list. This ensures that you get people who are active as you will follow people who followed that account in the past few hours. You can repeat this with the same accounts every few hours as well, as you'll always be getting new people. I wouldn't say it's super effective, but it gets the job done over time.
Method 2: Mass following people, but in a more targeted way. I think this way is significantly more effective, but it requires more work as well. The same downsides of mass following apply here, be careful to not do it too often so you don't get your twitter account flagged.
To do this, you find a twitter account that is fairly active. I'll use
https://twitter.com/SimpleGain as an example. Find a more recent post, no older than 2 hours, and open the twitter post up.
The area I circled here is the key. When you click on either Retweets or Likes from within a twitter post, you can see the 20-25 most recent retweets/likes. Following these people is a much better method to gaining followers as you already know the majority of these people are looking for mutual followers already. Again, locating a bunch of twitter accounts and doing this once or twice a day to each twitter account gets you a lot of mutual followers.
Method 3: Retweeting other accounts that are attempting to gain mutual followers.This method is very effective but the downside of it is that it really bogs down your twitter feed which may result in losing followers as well. You definitely gain more than you lose, but you lose your follower loyalty by posting garbage non stop. I'd say it's worth doing, but mostly only off and on.
To do this you will need to find some twitter accounts that post things constantly, I'll use
https://twitter.com/SimpleGain again as an example. This account posts something every 15 to 20 minutes, which makes it perfect for retweeting constantly. Once you locate other twitter accounts such as this, you simply retweet/like their latest tweet, and you can do this every 30 minutes or so. People tend to follow people who retweet or like popular posts, especially posts that are specifically designed to get more followers. By doing this you get on their retweet/like list and people will follow you from there. With this method, be sure to check your new followers constantly and use the extension to follow them to keep track. Again, this method is very effective. I believe there is some method to automating their tweets in some way, but I haven't looked into that yet so I can't comment further.
Method 3a: Making your own tweets that are attempting to gain mutual followers.Just as a side note, you can make your own tweets. I mentioned automating your tweets, but there's nothing stopping you from making your own posts at the same times every day. The more you post, the more you followers you get. This one is kind of a no brainer, but it takes significantly more time to get rolling if you have no followers to begin with. Be sure to use similar hashtags as the example twitter account here
https://twitter.com/SimpleGain. The key here is to be consistent, post the same times every day or constantly throughout the day at set intervals. It takes a lot more effort, but you'll get better results as well. I don't really enjoy doing this method but that may change once I learn how to automate tweets.
Method 4: Mass liking popular tweets and retweets.This is my favorite method and the one I use most regularly to gain followers. Using
https://twitter.com/SimpleGain as an example again, simply find 10-20 different accounts that have similar posting habits and like every post they made. The effectiveness of this is pretty low with posts that are over two hours old, but anything before that is great. Every time I do it I get about 5-10 followers on average, and it takes very little time at all. With this method you have to remember to check your followers pretty frequently and again, use the extension to follow them to keep track of dirty rotten unfollowers.
Additional InformationThere are some obvious things about your twitter account that are important too.
You need a profile picture and you need a background picture. Try to get something that's not horrible looking and spend a little time on it. It matters to a lot of people. It's better to have something than nothing at all.
You need a bio as well. If you don't feel like writing anything, just fill it with hashtags. #followback #followme #bounty #bitcointalk are all pretty valuable ones in my opinion, but the sky is the limit. You want your bio to either draw people in as part of the search engine or because people are interested in you.
Your pinned tweet is important as well, you can extract followers that way as well. If you are using it for something else or a bounty, that's fine too. Just be sure to have something there.
Does anyone else have any good methods for gaining followers on twitter? Please share them here, I'd be interested in trying them out! Additionally, if you know how to automate tweets that would be valuable information as well, some of the accounts that have the biggest growth are purely just tweeting every 20 minutes non stop forever.
