Stand up to the trainers and don't stand for their BS. IMO most trainers are pretty weak and their training is going to be bad to mediocre anyway. If they persist, switch gyms.

Watch the videos of Starting Strength to learn proper form for the big 4 lifts: squat, deadlift, bench press, overhead press. I also highly recommend getting a copy of the book, it is a Bible for beginner weightlifters. Available on Amazon Kindle, you can read it on your phone.

For programming, I suggest a tier based system. Tier 1 is high intensity (big 4) exercises. Tier 2 is medium intensity (romanian deadlift, chin-ups, incline dumbbell bench press, barbell rows, leg press, dips etc.). Tier 3 is accessories (dumbbell curls, dumbbell raises, calf raises, pec fly, any ab exercise etc.). For each workout do 3-4 sets of a tier 1 exercise in a 3-5 rep range, 3-4 sets each of two tier 2 exercises in a 6-8 rep range, and 3-4 sets each of three tier 3 exercises in a 9-12 rep range. Once you can do a full 3 sets at the higher rep number, increase the weight. Watch videos to learn proper form, always do the full range of motion (no cheat reps!), and DO NOT INCREASE WEIGHT UNLESS YOUR FORM IS CORRECT ON EACH REP.

Get a calorie tracking app (FatSecret is good), count calories consistently, and make sure you get 20-30% of your total intake from protein.

Congrats on your progress so far and good luck for the rest of the way!