Here's a weight training routine for the back that includes exercises targeting different areas of the back:

1. Deadlifts: Deadlifts are a compound exercise that primarily target the lower back, but they also engage the entire back. Start with a warm-up set or two, then perform 3-4 working sets of 8-12 repetitions.
2. Bent-over Rows: This exercise targets the middle and upper back, as well as the rear shoulders. Hold a barbell with an overhand grip, bend forward at the hips while keeping your back straight, and pull the barbell towards your abdomen. Perform 3-4 sets of 8-12 reps.
3. Pull-Ups/Chin-Ups: These exercises target the upper back, specifically the lats. Use an overhand grip for pull-ups or an underhand grip for chin-ups. Perform as many reps as you can for 3-4 sets. If you can't do full pull-ups or chin-ups yet, you can use an assisted pull-up machine or resistance bands for assistance.
4. Lat Pulldowns: Lat pulldowns are similar to pull-ups and chin-ups but allow you to adjust the resistance. Sit at a lat pulldown machine, grasp the bar with a wide grip, and pull it down towards your chest while leaning slightly back. Perform 3-4 sets of 8-12 reps.
5. Seated Cable Rows: This exercise targets the middle back and rear shoulders. Sit at a cable row machine with your knees slightly bent, grab the handle with a neutral grip, and pull it towards your abdomen while keeping your back straight. Perform 3-4 sets of 8-12 reps.
6. Dumbbell Rows: Dumbbell rows are another exercise that targets the middle back and rear shoulders. Place one knee and one hand on a bench, hold a dumbbell in your other hand, and pull it towards your abdomen. Perform 3-4 sets of 8-12 reps for each arm.

Remember to use proper form, select weights that challenge you but allow you to maintain good technique, and gradually increase the weight as you get stronger. It's also important to warm up before starting the routine and cool down afterward. If you're new to weight training, consider working with a qualified fitness professional to ensure you're performing the exercises correctly and safely.