(1) Follow as many people as you can with similar interests (e.g., biology, stats, disease ecology, etc.), and they will likely follow you back. This will increase the number of followers you have quickly. Just make sure that your little blurb reflects who you are and what you do well- so they can quickly evaluate if they have similar interests. For example, mine says:
Post-doc at Michigan State University, Disease Population & Community Ecologist, Herpetologist. Check out my website: http://grazielladirenzo.weebly.com/
Try using lots of key words that other people can identify with- and will say "oh yea! I like population ecology".
Personal strategy: The way I look for followers is I go to someone else's twitter page that I know has a lot of followers (for example Karen R. Lips @kwren88), and I just start following her followers or the people that follow her.
Go to hashtags or Society journals twitter pages that are interesting to you- and start following people that have similar interested.
(2) When you tweet use A LOT of hashtags and use hashtags that people will look at. You are now probably wondering, "How do I know what are the most popular hashtags?". There's a website for that!
Go to that website, and lets say you are going to tweet about science- so put in "science" into the search bar and scroll all the way down to the network map. This map shows you how frequently people visit those hashtags, and give you a sense of "will by tweet get swamped out by all the other people tweeting about science? or will it be visible for a little while?"
(3) Plan your tweets ahead of time (check out this blog)- you want the most people as possible to see your tweets and to get your name out there. According to that blog Noon to 1pm is the best time to tweet. I think there is a way to schedule tweets and have them come up at a certain time if you are unavailable.
(4) Tweet about websites (that help early career students, advice on how to get a job, mental health problems in academia, how to put together a good presentation, remembering to be color-blind friendly, how to deliver a good seminar, etc.), articles (your newest publications, exciting publications you like, try reading a paper a day for the next year and follow #365papers), and science events you attend (i.e., Euring, ESA, workshops, etc.). Tag people associated with those websites, articles, and events.
(5) Not so much a strategy as a fledgling twitterer: Retweeting will help your followers see things that you liked from someone else you follow. This may not necessarily help you gain followers when you are just starting, but it helps spread interesting articles, ideas, websites that you have found to your followers. The value of retweeting increases as you gain followers (I think)- but you can probably dispute that.
So take home message: Start following a lot of people (even if you don't know them), start tweeting a lot (websites and articles), use lots of hashtags, and tweet at appropriate times.
UPDATE: I had a friend recently ask me, "I have a personal twitter, and I don't really want to give that up. What are the benefits of having a scientific twitter?"
Here is my response:
(1) When I really started using twitter (i.e., following others)- I swear that I got at least 4 peer-review journals ask me to review papers. I think I got 2 more quickly after that, but I had to decline because I was already doing too much. I don't know if me being more active on twitter was correlated with the increased exposure of my name and my science- but I definitely think it got my name out there.
(2) For someone that is shy and has a hard time approaching people at conferences, using twitter has made it is easier for me to talk to people (like, "oh- i really liked your post on X").
(3) From the items I was tweeting about (i.e., being color-blind friendly with your slides, how to give a good presentation, open source science, etc.), I was invited talk to the Mozilla Science Fellows and to the Kuris/Lafferty/Torchin parasitology group at UCSB.
(4) Lastly- you get to see cool pictures of what other people are doing, people post papers a lot (some of which I may never have stumbled on), and it is a great way to see exciting science at conferences you can not attend by following their hashtags (i.e., #ESA2017; for the Ecological Society of America meeting in 2017).
Again- I think there are a lot of advantages to using twitter as a scientist- but it is ultimately up to you to decide if it worth the time and energy.