You need to build your community from the get-go. Start with your beta testers, creating a community where they talk to each other about issues and come to you with ideas, suggestions and bugs. Encourage them to do this, no matter the issue, and respond quickly.
If you develop a community like this from the start, it's very easy to get your users to give advice on improvements and the like. You'll quickly see the beta users who are very involved, and ask them privately for suggestions of other beta users.
Beta users WANT to provide feedback, and they will provide some of the must useful feedback you can get, so long as they see you responding and taking into consideration at least some of their ideas. If you start this in beta, you'll build a community of users who want to help you succeed and want to help you improve your product, for the long haul.
I've been involved in everything from concept research (Harley-Davidson, Hallmark) to formal usability testing (United, NatGeo) to intercept surveys (Purina, SC Johnson) and so on.
The fact that you are interested in understanding your users' needs puts you ahead of the game already.
As usual, the way to best understand these needs depends heavily on your research objectives, budget, and time frame. There is no one right answer.
So, in the spirit of the Olympics, here are some bronze, silver and gold options.
Bronze #1: Add a button in the app that says "How are we doing?" or "Help us improve..." or similar and link to a simple email form. Language is everything here. If it just says "Contact Us" it will be ignored by everyone except those who have a beef.
Bronze #2: Send an email to your core user base -- the group within the group  if you will -- asking them to a couple of well-worded questions. Again, wording the questions is a critical art form that will ensure you're getting "real" answers that you can tie back to addressing your research objectives. Your questions must be clear, focused, and of the appropriate complexity.
Silver #1: Run an intercept survey on your website or if possible within your app. This will typically "cost" more as you'll likely want to spend time planning the survey questions, determining how you are going to screen prospective participants, and deciding how you'll report the findings and recommendations. Depending on the complexity of the survey and number of participants, you may find it worthwhile to engage a 3rd party, which range from simple self-service tools to full-blown research consultancies.
Silver #2: Perform a "guerrilla usability test" or "hallway test." Just Google it. Short version is it's a "discount" way to do a usability test, and often provides a much better value (cost-to-findings ratio) than a full-blown usability study.
Gold #1: Do some concept research. Essentially focus group where you're simultaneously trying to vet concepts before you invest too heavily in building them, and also to solicit ideas from participants. I've seen concept research done with everything from tissue sketches to fully developed products and everything in between (mood boards, functional prototypes, etc).
Gold #2: Perform a full-blown moderated usability study. As with intercept surveys, there are ways to do this for more or less time / money, but usually you'd want to work with a professional as findings from an improperly run usability study can do more harm than good. Unlike intercept surveys, usability studies are more qualitative so you get much richer texture from a much smaller group of people.
Those are really just the tip of the iceberg / knee-jerk examples. If you really wanted to go full-bore with the user-centered design approach you'll start getting into areas more similar to anthropology including ethnographic studies, contextual inquiry, and customer experience modeling and the like.
That said, hope these examples help you to get going in the right direction(s).
P.S. Feel free to reach out to me here if you'd like to talk in more detail about your particular research needs... https://clarity.fm/toddlevy
In my experience, there are two schools of thought. You might not like my answer -
1. Ask them
Take the time to bring them into your office and watch them surf and keep it open minded. Or hire an online video review company that takes videos and screengrabs of the whole thing. It's the advanced technology that a couple businesses provide at a fraction of the cost from only a couple years ago. The trouble is, people don't know what they want. They need it pried out of their minds - seduced out of their emotions. Wyatt Jenkins of Shutterstock is a master at understanding the minds of the consumer - he has written about this some (and about philosophy #2 even more) - but in my opinion, experts who can deliver results through extensive surveys like this are very rare and expensive.
2. Don't ask them (or just ask them by whether they part with their money)
Malcolm Gladwell explains it concisely I think when he discusses spaghetti sauce. Most of the times, your customers do not know what they want. The goal here is to provide your customers with many opportunities in many places to provide their feedback. And then like 37Signals says in their super business guides, just delete their feedback. Ha! Although that's kind of out of context -- the point they are making is that if the customers demand something over and over, you won't miss their demands because they will become repetitive. And then you must deliver and wow them.
I side with philosophy number two because it is more constructive with managing risk and sides well with iterative development. In my experience, focusing on key things, key functions, and doing them better than anyone else is by far the most important. The modern way to do it, is to keep the qualitative feedback from the customer out of it, and just test them by split testing your applications. Then you'll get the "money where your mouth" feedback is, because the revenue and conversions will be driving your decision making...and thus your app's evolution.
On the other side, if you are a boutique brand, and have plans for world domination, then perhaps you should take the chance on an extensive hail mary R&D project. I don't like that idea because of the #1 rule of investing - security.