Followers vs. Engaged Followers
At the outset, it is important to understand that simply increasing the number of followers one has on Twitter might not accomplish what the blogger is likely trying to accomplish (i.e., driving more traffic to one's blog). If you are interested in using Twitter to promote your blog, you are going to be far more interested in attracting engaged followers (i.e., those who will interact with your content) than followers in general.
Suppose you currently have 100 followers, and we are able to get you to 200 without much effort. That might sound pretty good; however, you might not notice much of a difference in traffic to your blog unless many of those new followers are engaging with your content (e.g., liking it, retweeting it, responding to it). So while it is tempting to focus mainly on one's follower count, the quality of one's followers in terms of engagement is more important than the quantity. But because it is almost impossible to predict who will engage with your content before they are following you, we have to start somewhere.
Selecting the Appropriate Metric
Even when it comes to engaged followers, not all types of engagement are equal. If your goal is to drive more traffic to your blog, likes and retweets are nice, but what you are really after is clicks. Your tweets contain links to a blog post you have written, and you want people who see your content on Twitter to click those links so they end up on your blog. While I would not ignore likes and retweets, I tend to be far more interested in clicks on those links you are sharing.
Prioritizing Your Twitter Interactions
When others interact with your content on Twitter, it pays to make interacting with them a priority. Most people on Twitter who see your content will ignore it. That is the nature of Twitter. Those who do not ignore it but interact with it are those with whom you should be interacting. This does not mean you should go out of your way to respond when someone says something stupid (which they will). But assuming you have limited time, I would recommend that you allocate some portion of it to interacting with those who are interacting with your content in a meaningful way.
Frequency and Time Zones
Consider how often you are tweeting and when you are tweeting. I recommend using a service that will allow you to schedule your tweets in advance and automatically send them at times you set up. This will allow you to tweet when your followers are on Twitter rather than only when you are on Twitter. I use Buffer for this, but I have been impressed with what dlvr.it offers too (especially considering what it will let you do for free).
How Often Do You Tweet?
If you want to use Twitter to drive traffic to your blog and you are trying to increase your followers, I recommend that you tweet somewhere between 3 and 15 times a day, 7 days a week. The minimum of 3 tweets is based on the understanding that you are trying to expand your number of followers (which means that tweeting more is usually preferable) and evidence of diminishing returns after 3-4 tweets (i.e., user engagement per tweet tends to drop after the third or fourth tweet). The maximum of 15 is based on recommendations that more than this tends to annoy others and almost certainly involves diminishing returns.
I recommend setting up a reliable system for tweeting 3-4 times a day and doing that for a couple weeks. Then I might increase my frequency to 5-6 times a day and see what sort of difference I noticed, if any. My personal bias has always been toward trying to maximize efficiency. That means if I can accomplish the same thing with 3-4 tweets/day as I can with 5-6, I'll stick with the 3-4. That said, I believe an argument can be made that tweeting more often may be helpful when trying to grow one's follower count, especially if it is done with an eye toward exposing people in different time zones to your content.
When Do You Tweet?
Since you are after growth, you need to spread your tweets out and think about hitting different time zones. This is where using one of the services I mentioned above becomes so helpful. If you only tweet when you are on Twitter, you are missing everyone around the world who is sleeping, working, or whatever else when you are on Twitter. Timing your tweets to go out when most of your followers are on Twitter is a good place to start; however, I have found that I tend to get the most blog traffic from tweets that I send out at off-peak times (e.g., 2-3am). This is sometimes referred to as the "late night infomercial effect," and it has been widely documented. When there is less content available, yours will receive more attention.
The more you plan to tweet per day, the more important it becomes that you spread your tweets out. Ideally, you should tweet no more often than once in an hour. This gets you the most potential benefit per tweet and prevents you from annoying your followers. I wouldn't worry too much about tweeting more often than this when you are interacting with followers, but I would refrain from posting links to your blog more often than this.
What Do You Tweet?
Since you are trying to drive traffic to your blog, you will want to tweet every blog post you write, and you will want to do so more than once. Again, think about time zones. When I say that you will want to tweet every blog post more than once, I do not mean that you should tweet exactly the same thing each time (Twitter does not like this). Ideally, you should vary it up. Maybe the first tweet is just the title of the post with the link and two hashtags. The second tweet could be a quote from the post with the link and two different hashtags. The third tweet could be a re-worded version of the title with an image and two hashtags. You get the idea.
What's the deal with the "two hashtags," you might ask? You want to make sure you are using hashtags on your tweets because this makes them more likely to be seen. But more is not necessarily better when it comes to hashtags. There is some evidence that tweets with two hashtags perform better than those with more than that. I think this probably depends on what one is trying to accomplish, but I rarely use more than two.
Even though you are trying to drive traffic to your blog, it is a mistake to tweet only your own content. If you want to grow your followers, content curation is the way to go. For every link to your blog you tweet, try to tweet at least 1-2 that do not contain links to your blog. Some of these can be tweets with no links, but many should be links to others' content. Not only does this provide value to your followers, but it may get the attention of some of those whose content you are promoting.
Social Media Managers
Crowdfire
Crowdfire is a popular social media manager with a free plan that might be worth checking out, especially for those who have fewer than 500 followers and are looking to grow their numbers quickly. Before you mess with Crowdfire, though, it is important to understand how it works and recognize some important downsides of the model it uses. This model is by no means unique to Crowdfire, so you'll likely encounter it in similar services too.
Once you set up Crowdfire, it will recommend that you follow a batch of Twitter accounts every day. It has been a while since I have used it, so I cannot recall the limit on the free account but let's say it is something like 25. It will present you with a list of these accounts, selected for you based on the information you entered when you set it up. Most will probably be at least somewhat relevant if you set it up correctly, but some probably won't. It will be up to you to decide how selective you want to be in terms of who you follow, but the model is based on the idea that you will follow most of those recommended. Crowdfire will also recommend that you unfollow a batch of accounts every day. If you have ever noticed that a handful of people follow you one day and then unfollow you the next, odds are very good that they are using Crowdfire or a competitor.
This model is based on some old data indicating that when you follow someone on Twitter, there is something like a 20-30% chance that they will follow you. So what Crowdfire is doing is helping you follow one batch of people in the hope that they will follow you. It will then encourage you to unfollow those who do not follow you within a certain time frame. It can be effective in getting more people to follow you, although you will almost certainly discover that many of the people you end up following are not worth following.
Having tried Crowdfire myself, I cannot recommend it because it runs counter to why I use Twitter. I follow people because I find value in what they tweet. It does not matter whether they follow me back. With Crowdfire, I ended up following lots of atheists who contributed little more than tribalistic nonsense and name-calling. It did not matter if they followed me because I quickly unfollowed them as soon as I saw how they were behaving. Crowdfire and similar services can be an effective way to increase one's overall number of followers, but one needs to recognize the downsides of how their model works. You will probably end up following people you don't want to follow and being followed by people who do not engage with your content.
Commun.it
Commun.it is another social media manager with some similarities to Crowdfire (e.g., it will recommend following some accounts and unfollowing others) but a different emphasis that makes it interesting. Their free plan is worth a try if you are not annoyed by the constant pop-up ads trying to get you to use their paid plans or the chaotic user interface. I tried one of these paid plans for a month and was not impressed. Stick with the free version.
Unlike Crowdfire, Commun.it emphasizes identifying and interacting with your most engaged followers. It is going to suggest that you unfollow those who are not engaging with your content and help you identify them. But beyond recommending people to follow and unfollow, it helps you see who is interacting with your content and makes it somewhat easier to interact with them. The free version will give you something like 20 engagements per day, and this has been more than enough for me. I do not use Commun.it more than once every week or two, but I have found it useful.
The Follower to Following Ratio
The last thing I'd like to mention is the ratio of how many followers you have to how many people you are following. Do not be overly obsessed with this number, but keep an eye on it. Ideally, you will end up with more followers than people you are following. But initially, it is fine for these numbers to be close together (i.e., a 1:1 ratio). What you don't want is to end up following far more people than the number following you since some people use that as evidence that you are not worth following.
Personally, I follow those whose content I find valuable and who do not repeatedly annoy me. I tend to unfollow at least a few people every day because of the annoyance factor (way too much petty tribalism and name-calling), and I follow people at a slower rate. As long as you are actively trying to grow your follower count, I think that having a ratio close to 1:1 will probably make good sense.