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 [1] 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).
TL
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
[1] http://www.shirky.com/writings/herecomeseverybody/group_enemy.html
