So a personal trainer may actually be good for you. I had the issue at first when I first started going back to the gym (I did power lifting and sports in highschool then stopped caring from 18 to 25) and I had forgotten everything was had anxiety going into the gym (that's where my girlfriend came in and helped me, she's actually a personal trainer herself so I'm very fortunate).
From what I've read and even experienced myself I would suggest both weight training and cardio. In that order. Your body has two, I'll call them reservoirs, of energy it can use. The energy you get from the protein you eat during the day and then that of whats stored in your body in your fat cells. Now cardio will burn through most but not all of your protein intake if you do it first and it also doesn't use it as efficiently as weight training (protein is a huge factor in muscle growth). So by weight training first you use your protein to help fuel your body and grow your muscles and by the time you get to cardio your body is ready to use the other energy in your fat cells to burn through to keep you going during that.
Also yes your diet is a HUGE VERY HUGE factor. You can workout 7 days a week for hours on end but still see very little improvement. Your diet and what you eat is 70% of the solution. You can start tracking calories with an app on your phone that lets you scan the products barcode to get it's information. You also need to stay away from huge amounts of sodium and fats. I only cheat on my meals Friday nights and Saturday all day. Every other day is healthy and meal prepped for the week.
Your teach is smart and right when he/she makes you do that. Being able to do one amount of weight at one rep isn't going to help. Now doing a lighter amount for a huge amount of reps is what will.
For your muscles to grow you need to tear them apart which is what weight training does and then allow them to recover and repair themselves. And repeated actions will help greatly with that.
For toning your body and defining your muscles you always want to do lower weights for high amount of reps (for instance on shoulders I do shoulder press, lateral raises, side raises at 4 sets x 12 reps at 20 pounds).
For growing muscles you want to go from lower weight with high reps to higher weight with low reps (example shoulder press: 12 reps at 20 pounds, 10 at 25, 8 at 27.5, 6 at 30) and when you start getting stronger you keep raising the weight for each set.
Now your shoulders hurting from deadlifting shouldn't be happening and could be due to form. Deadlifts are for your legs and glutes. That's what you should be using to lift the weight, nothing else. Your hands and arms are there only for you to hold the weight to lift it. Depending on how many reps though your arms/shoulders can be tired from holding onto the weight for so long.