Strength training is where its at for menopause to help build strong bones.
And integrating cardio with that is a good idea too of course.
If the induction includes recommendations for exercises, I'd say its likely to incorporate both of these things regardless of how clued-up or clue-less the trainer is about menopausal health.
My PT also recommends core/mobility work like yoga or pilates as I'm heading into my menopausal years, but I'm a bit lax with these TBH because it's not really my bag.
In your OP you talk about wanting to "improve strength, flexibility, stamina, and of course lose weight". Is that your order of priority? If so, I think its great. I see so many people at my gym who're over-focused on losing weight at the expense of overall health. But overall health is much better achieved through building strength and bone density and flexibility and lung function and heart health etc. These things combined with cardio and a good diet will help you lose weight eventually but be patient and prioritise health.
Strength training and core building are also your best change to build a good frame/posture which will help you carry weight better anyway. So you may look slimmer even if you've not lost weight. And when you do lose weight, you'll lose it well.
I should add that the 20-year old at the gym won't, or shouldn't be, clueless. In my experience, the trainers are all very knowledgeable and the young ones who're fresh out of training/university are often especially bubbling with knowledge - its very fresh for them.
Around here PTs range from £20 per hour to £45.
> The £20 is for one-on-one martial arts training.
> The trainers at my gym charge £25 and do the sessions in the gym with you.
> There's a guy who charges for £30 for a session in his garage which he's kitted out as a small gym.
> There's a woman who specialises in women's training and strength training. She's got a big following of menopausal ladies and charges £45 for a one-on-one in a small gym that she rents from our local rugby club.