Agree it's never a bad idea to go to a *good* running store and get fitted. I heard Fleet Feet in Silver Spring is good for that; I'm sure there are others.
There is some quiz at Road Runner (or maybe some other site?) that will ask a bunch of questions and give you a short list of recommended shoes. FWIW, I took the quiz after my last fitting, and got back a list of 5-6 shoes that matched almost exactly what the running store had recommended.
But honestly, while I'm sure some people need just the perfect shoe, I think most of us who aren't truly competitive runners and don't have specific issues would be fine in most shoes.
Source: Been running regularly but non-competitively, about 15-20 miles per week on average, for 10+ years. I buy shoes whenever I find them on deep discount, so I've tried several brands. Really like Brooks, Asics, and (surprisingly) Hoka. Have one pair of Mizuno that I really want to like based on recommendations, and I try them every few months, but I never enjoy them. Tried New Balance and Nike in the past, and they were OK for me, but noting special. Used to like Saucony a lot when I was younger, but then got turned off after a couple pair that felt weird.
I suppose if you're just looking for one good starter pair, I'd suggest Brooks or Asics, but that's mainly because I like them for me. Don't get too hung up on a particular brand, because most of us plodders can probably use almost any brand/model without any problem. Get a decent pair (not bargain basement) because you don't want to be turned off of running by bad shoes, or have an excuse not to run.
Good luck! Getting a new pair of good shoes is really fun IMHO.