A few days ago, I had the opportunity of seeing the reasons people had given for wanting personal training - in a commercial gym setting. The thing that struck me most, was how incredibly vague to reasons were! I see people, well-intentioned as they may be, not achieving anything worthwhile because they simply have not taken the time to give it some thought. It is likely that they are bowing to the subliminal pressure of what it expected - lose weight, tone up, work on your core... etc. They might be admirable pursuits, but they are so prodigiously vague, they become meaningless and lead nowhere.
If you are going to the trouble of hiring a trainer to work with you alone, help them out and at least be specific about what it is that you want.
Be specific about what you want to achieve (your goals), and be specific about what you do, not only on how to achieve them (exercise choices), but equally how you action them (alignment, postural technique).
Many don't achieve anything noteworthy, not even remotely in the region of what they want to, simply because they have not defined their goals. "Improving strength" or "becoming flexible" or "toning up" or "getting fitter" or "working on your core" - is so incredibly vague it's actually meaningless.
How are you measuring strength, flexibility, tone, fitness or core? Where do you rate on that specific scale? Where do you want to be on that specific scale? How long do envisage it will take you to get from where you are to where you want to be? What specific plan do you have that will ensure you make those changes in that timescale? Do you know how to properly align yourself, and how to properly engage in the right places to ensure the exercise does what it's supposed to do?
If you can't answer these questions - all of them - you simply do not know where you are going, and then like the caterpillar in Alice In Wonderland says, it doesn't matter which road you choose if the destination is irrelevant. The journey itself becomes irrelevant. Question everything - and have answers.
Be specific. Or you will never truly make progress.