Congratulations on starting to lift!
Krista (Stumptuous) is a fabulous resource for women interested in iron! Sally Moss (the Gubernatrix) is the other major female resource. And congrats on hiring a PT to help you understand how a lift should feel.
While soreness (DOMS) is a funny thing, I find that I am rarely sore from lifting and I get great joy from it. If you get into it, you will find that your scale is going up, but your clothes are fitting better (or differently - but in a good way). I also find that lifting helped with the hormonal stresses of being a middle aged woman.
If you really enjoy it, DO use a gym membership or you will find yourself regularly wanting to expand your weight collection at home. I'm to the point where most of workout is done with a barbell and weight plates; it took me about 2 years to get to that point, but I'm betting it doesn't take you that long. (I was held back by a trainer).
At the moment I'm training for my first power lifting competition next month. My squat is sad (but I just had my hip replaced a year ago)... but my deadlift and bench are acceptable. Especially for a 60 year old female.