EDIT: I have since amended my Instagram strategy as IG itself has changed. Keeping this post up for historical reference.
I saw that my friend had great photos, but not really the number of followers to match. She had almost 1000 photos but just under 2000 followers. These were all top quality photos; she's a lifestyle/fashion blogger, so I offered to help out with her account. It's good experience for me to manage different accounts after all as a Social Media Manager.
I should also mention that my friend works full time in finance, and she's studying, so it's perfectly reasonable that she doesn't have that much time to spend on Instagram.
Upon inspection of her account, it became apparent why she didn't have the number of followers I expected.
She was following the wrong people
She had just under 2000 followers and only following 300. Most of those 300 were top names; people that are unlikely to engage with you, like Justin Bieber and Adele. What you really need to build followers, is to follow active people on Instagram. People that engage with you and will follow back. There's no real reason to not follow her account - the shots are beautiful, not selling anything and not offensive.
It was a slow start, but I had to start following some active people. How did I do that?
I looked on the recommended news feed for people with a fairly even ratio (followers:following). People I know are likely to follow back.
When I found these even ratio people, I looked at who liked their photos. I added and engaged with those with even ratios. This slowly increased the percentage of well engaged people in my news feed, so I could then start working off of that.
I also followed people I'm following from my own personal account - people with interesting photos that are engaging, and wouldn't offend my friend when their pics pop up on news feed (I did follow the naked assman, and she had quite a shock when she checked her instagram feed that morning.)
After a few days of this (it was a slow start), I then added and engaged with people systematically at the top of the news feed. People that have posted, and people that have liked their post. It was a revelation to see how quickly these people added you back.
I have found certain communities better to follow. Photographers, food pic people, and Turkish Instagrammers (the ones I have found anyway) are great at following back, and also are active and have interesting photos. Lifestyle bloggers in general seem to enjoy having more followers than follows. Maybe they are just very time poor, who knows.
I found also that as her following:follower ratio improved (ie it was becoming more level), people were more inclined to follow back. It was still a struggle because there were quite a lot of people unfollowing the account as they didn't get a follow back or engagement previously.
Within 2 weeks, she had gained 20% more Instagram followers, with over 2400 followers.