what's with you?
This situation is one that isn't going to have a definitive answer, but if you get enough of us providing input you'll probably get a good cross section of answers.
I agree that seeing a few other SPs will help you determine if it was 'real' or not. DON'T go to the SPs at your new infatuations' place of business. IF you end up together that could be a sore spot for her later on, I think.
I"ve had 'just business' encounters and I've had ones where there is a connection. Usually the connection is of a 'friends' nature or possibly a 'friends with benefits' type mainly because the SP wants to keep some kind of boundary in place. There have been cases where the connection led to me dating a SP and it became serious. In these cases the girl was in transition to a regular job and I think that had a lot to do with their willingness to jump into it. The SPs that kept it to a certain level were doing it because they wanted to keep their business going.
We know that you are new to the game, but we don't know about the SP. She could be an old hand, or brand new, full time or part time. That would probably effect how she is going to look at this. The SPs I serioulsy dated were usually doing it "only for a certain time period" or in one case I was her first client, but all were phasing out of the industry.
There is such a wide range of personalities and situations that it's very hard to generalize.
Your desire to take this cautiously is not the way I approach this stuff. However, when there are no indications that are obvious, such as the SP asking you out directly, a 'safe' way to broach the subject is if afer you're getting ready to go, the SP asks you what you are doing later Most of them are just making chitchat, but if you mention that you have no plans, and ask what they are up to that leaves an opening which doesn't put anyone on the line. It's still an innocent question and you can find out where the person is at without embarassing anyone.