It doesn't sound like the best routine to me. I just add 5 pounds whenever I can do all 24 reps of a 3 set 8 rep workout. There's some truth to goodness in staying at the same weight for a couple workouts to keep form good but not an entire month. And taking a week or two off every year is all you need for 'recuperating'.
It sounds like you're listening to a goofball. If you want to get big fast just do lots of compound lifts and wait 72-96 hours between repeating them.
What I do:
Deadlift, bench press, overhead press, dumbbell shrug and side raise
wait 1 day
Barbell bent row, preacher curl, dip, stomach crunch
wait 1 day
This is basically a full body routine with emphasis on compound lifts for size and strength. Don't put lots of time into tiny isolation movements like the preacher curl or leg curls until a year of training or when you're satisfied with your frame size. They will slow down your progress getting huge, and will not be as significant as if you start when you're at a size that you would like to stay at. Compound lifts will increase your body size as well as your muscle mass, and toning will become less visible as they shrink back into the larger muscles.
The only thing that the trainer you're doing is suggesting is that you slowly progress in an overly safe manner that turns into toning that shrinks plus some cardio and more likely a lot more form safety than you need.
Just follow your instinct and train as heavy as you want. If the weight/form starts feeling unsafe don't add any for a few workouts and you'll be fine.
As above, x3 workouts a week with squats, deadlifts, benching & a type of press. These should be the core of your workout. Squats twice a week.
You then add assistance work to each workout eg bicep curls, Side raises ect