I would ask them what happened today and go from there. If I was 30 mins late for work, my boss would be mad and as much as they are doing a service, you ARE paying them. If they have a good reason (like there was an accident and nobody could get to the facility), then I'd let it go. If not, I'd be asking about compensations and assurances that it won't happen again. If they are not apologetic, then I would be more upset than if they said sorry. That's a long time to wait, especially if you did not know when someone would arrive. In the winter or in inclement weather, it would be an even bigger deal.
When I had DD in daycare, I had to leave within a very specific timeframe or I'd be late for work myself. I chose a center because it was expected that someone - even if it wasn't "her" teacher - would be there to take the children and care for them during their posted hours. Had they regularly not been open on time, I would have looked for other care because otherwise I would have been fired myself. People who drop their kids off at 6:30AM probably chose that center for the early hours. If they can't do the early hours, then they need to change their hours. And as you said, every place I have ever had a kid (aftercare or daycare) charged you if you were late, and if you were late often, you got kicked out.
SO....short version - ask at pickup what the deal was and decide what their answer means to you. You just started, but you still deserve to know what kind of service you are getting.
And while you are new, don't let that stop you from asking legit questions about your child's care. I'd rather be "that parent" than accept something I don't agree with or stay with a bad center. There's a difference in being assertive and being craptastic.