From over 30 years of experience in this.
Number 1 thing - get specs. Make sure they are written down, but not to the level of detail which will make your internal customers eyes glaze over. One problem I have seen is that non-programmers have no idea of how their little tiny change can screw up a project, and also how some things might be easy to change. Getting them to commit will be like pulling teeth, but an investment in listening to them, telling them what you heard, and repeating it, will pay off big time.
Numero two-o - as TriPolar said, have something to show them, and have it quickly. The hard part of the code will likely be implementing stuff which is pretty solid, stuff like the UI will be the things they want altered, since they will not know what they want until they see it, and sometimes not even then. The prototype is actually part of the spec, but helps in that your customers will understand better that you are making progress, and your team will see some real feedback.
Three. Understand the skillset of your team, and break down the job accordingly, not forgetting the people doing testing, who should be starting to define and write tests when the coding starts. Get estimates for completion from each. My heuristic was to double it and add 2 weeks, but you might not have that luxury. Have plenty of visible milestones to track progress - visible meaning a function completed, not something like “coding 90% done.” Make this very visible. When I was in grad school I did 90% of a quarters work in the week before our quarterly report was due. Use project management software if you must or if it gives you a warm, fuzzy feeling, but remember that this software doesn’t do a good job predicting debug or handling rework, and so you management might consider the end date it predicts the truth, and not a wildly optimistic guess.
Do code reviews. You probably shouldn’t attend, but get your best and most confident programmer to start. This will make it clear that getting ones code review is not a punishment, and also starts with the biggest plus for code reviews, which is the good programmers teaching the not so good programmers how to do it by example.
As for you, the three best things you can do is to understand how everything fits together, and make sure that your staff don’t make design decisions that locally optimize their part at the expense of the project; worry: that is, think about things that might go wrong and might be forgotten, and try to ensure that you’ll be okay if they do; and protect your team from one more change or from being taken away temporarily for “more important” jobs. Oh, and track completion dates and bugs, so you can do better next time. And if you must make a change in the middle, estimate the schedule impact, get on the agenda of a top person, and ask them for formal buy-in for the slip. They may tell you to work smarter, but if you have been delivering your first milestones on time they may actually rethink the oh-so-important change. If it is really that important, they will at least know a realistic delivery date, and if you manage to beat it the better for you.
Oh, and whatever you do, don’t overpromise. That is hard for a new project leader to get - until you can’t meet your promises. Underpromise and overdeliver is good advice.