While you certainly will become familiar with the brand/generic drug names as you work in the pharmacy, sometimes when you're just starting out, it's too much new information at once to pay attention to which names are equivalent (ie, Ambien is zolpidem). If you have the time to sit down and memorize the brand/generic of a few of your pharmacy's common drugs each week, it will help you. Eventually they'll be second nature. You should also know what the drug's general use is (antihypertensive, analgesic, antidepressant, etc-- you don't have to go in-depth). Knowledge of drug names and their general uses are required to pass the Walgreens pharmacy technician course.
Being a good pharmacy technician takes time. You'll eventually need to master everything from cashiering at the register, to inputting and filling, to dealing with insurance issues, to ordering stock from the warehouse. Just learn each skill set as you come to it as thoroughly as you can (take notes!) and then work on doing it more accurately and efficiently.