What works is the good guy who knows what he wants and goes after her. He calls, he asks you out, he pursues wholeheartedly. He’s emotionally available and he wants YOU. Would he initially love it if you made the first move? Probably. But then he wouldn’t. And you would hate it. No one wins when the alpha female pursues.
Bottom line: you have everything to gain and nothing to lose by letting him make the first moves. But there’s one more issue in the story you’ve told. I’m wondering why this guy who is supposed to be your trainer is “hitting on you.” Is he a guy who hits on women in the gym, using his job to choose his next sexual conquest? Or is he a good guy who is genuinely attracted to you but reluctant to make the first move because you’re a client? Here’s the real opportunity for you: to clarify his intentions so you can either move on or move forward. A suggested script for that might be this one:
“Thanks for helping me get started with my fitness evaluation; I appreciate your help! However, I couldn’t help notice that there seems to be a bit of flirting going on here. Am I imagining that, or are you feeling that as well?” If he says you’re imagining it, then ask him to please focus on your fitness process instead of asking so many personal questions (good boundaries).
But if he indicates that he is feeling the attraction then say something along these lines. “Well, I’m glad to hear I’m not just making it up. But if you’re interested, please call me later and ask me out; here’s my cell phone number.” Then let it go and re-focus on your workout. If he calls and asks you out and you hit it off, ask for a new trainer; otherwise it will be awkward. If he doesn’t call and ask you out, it will be awkward as well, so be ready to request a new trainer.
There is one other option here. Just do nothing. Re-direct the personal questions, focus on your workout, and let it go. If he’s really into you, he’ll figure out what to do and he will take action. If he’s not into you, or if he is but he’s not available, then you’re still better off because you didn’t take the lead.