I share @JDT 's opinions overall, but with a few tweaks
Senior software engineer
My personal first pick. Why? Because he'll basically be the glue between the view and the data. Good ones will typically also be able to mock up the database and the UI. This helps to clarify the requirements and identify road blocks early on. It's also a difficult to find one ...a good one.
Junior software engineer
Definitely not the first hire. You don't want a noob to mess up the foundations. Also, the senior should have a saying in who you hire as a junior. After all, they will be working closely together for quite some time.
Web Designer (UI)
Second/third. First, you want things to work, then you want to make it nice. Don't underestimate it though, because making the UI prototype early on will typically show that some features are required or should be altered. Also, it should not be underestimated. I've seen several apps where the front-end was more work than the back-end.
Second/third. At the company I work, we don't really have dedicated DB admins, it's part of developers job. I can only imagine a dedicated DB guy when there is some really heavy DB lifting. However, data is also the corner-stone of the app, so it should be done right.
The last. Once at least the prototype is ready.
A last word about senior/junior. It's more about competence than seniority. They are only very loosely related.
One more note
I don't know how it works for you guys, but where I work, most people are typically cross boundaries. There are the developers which do back-end and database, and the ones which do web UI and backend. Indeed, most of the work is about connecting the dots, not working on each aspect in isolation.