For a good, very concise overview check out Sports Fitness Advisor Tennis Training: http://www.sport-fitness-advisor.com...-training.html
It sounds like your trainer is working to get you a good foundation of strength, but trying to get in a little of actual muscle strengthening as well:
"Phase 1 - Foundational Strength
Perform the following tennis weight training routine 2 days a week for 6-8 weeks. It focuses on the major muscle groups and aims to strengthen the whole body. It should be performed during the off-season when you have a break from playing tennis entirely. Complete the exercises in order and try to leave at least a day or two between sessions.
You should also warm up with 10 minutes of light aerobic exercise before a tennis weight training session and cool down with 10 minutes of light exercise and some stretching exercises.
Load: 40-50% 1 repetition maximum
Rest Interval: 90 seconds
Dumbbell Squats or Lying Leg Presses (legs, glutes)
Dumbbell/Barbell Bench Presses or Push Ups (chest, triceps)
Back Extensions on Stability Ball (lower back)
Dumbbell Lunges (legs, glutes)
Single Arm Dumbbell Rows (upper back, biceps)
Crunches with Twist (abdominals)
Dumbbell Shoulder Presses or Machine Shoulder Presses (shoulders, triceps)
Standing Barbell Curls (biceps)
Standing Machine Calf Raises (calves)
Barbell Upright Rows (shoulders, trapezius)
You should also perform a rotator cuff and forearm program with light weights 2-3 days a week. This can be done easily at home and should take no more than 20 minutes to complete. Try to complete the program on separate days to your weight training sessions."
While you are overweight, this is a reasonable approach, as running will place a lot of pounding on your joints.
The 10 minutes on the treadmill sounds about right, but if your knees/hips are aching, you can cross train on a stair stepper or even a bike (although cycling markedly reduces the work because you don't have to support your body weight).
You probably already know it, but changing your diet is key - eliminating junk food, fast food and high calorie drinks is a great way to start.