Programming for strength training is based on evidence-based scientific principles of adaptation. Your training plan is written like a prescription to help you progress toward your goals, with every detail planned around the desired adaptation (whether that be muscle growth, muscle strength, fat loss, improved jumping power, etc).
If you are selecting a weight that allows you to do a LOT more reps than the program indicates, then you should increase the weight you select, not the reps you perform, as performing longer sets will result in a different adaptation. Read the instructions of your program carefully to make sure you are selecting appropriate weights for your goals.
For instance, performing 3 sets of 12 reps would not result in an equivalent adaptation to 6 sets of 6 reps, despite having the same overall volume (volume = sets x reps). 3 sets of 12 will help grow muscle mass better than it will strengthen the muscle tissue, while 6x6 will strengthen the muscle more effectively than it will grow the muscle size. These are two different types of muscle hypertrophy and we've learned through research that BOTH can occur in any rep range but are optimized within specific rep ranges.
It's important to know this going into your first strength program -- don't try to "cheat" the program or you'll only cheat yourself. Cutting rest intervals short and increasing reps to meet the volume because you're short on time is a sure way to self-sabotage the results you're seeking.