There's a ton of reasons one can have stones: diet, genetics, congenital deformities, flow (urodynamics), hydration, there are even some bacteria that can cause them...
I'm sorry my colleagues told you that there's only one type, and no way to tell. It's technically true with the data given, but...
Generally, a specialist visit isn't indication after the first stone, but if you want to see one, there's no reason why not. If you want the best result, you'll **** through a strainer until you pass what looks like sand. You'll take that to your urologist and they can send it to the lab.
Stones happen for multiple reasons (above), and while calcium are the most common, we can do better job of telling you what your risk of future stones are and/or what you can modify to reduce that risk of recurrence of we know what caused it, which we can't do just from knowing you have one.
You can try flomax, toradol (ketorolac) or one of its cousins, and push hydration. Other than that, I can't be more specific over the internet with the data given.
May I suggest going to your primary? They can likely do 99% of this and send you to a urologist or nephrologist as needed.
Now, please don't mistake this for comprehensive medical care, but it's pretty close to what I'd tell you if you're in my ER.
And I may not be a doctor, but I did stay in a holiday inn last night... Just kidding, I actually am one.