Wanted to confirm that not getting your withdrawal bleed immediately is totally normal. This is especially true during the first few months while your body is adjusting to the hormones. You may not have a withdrawal bleed at all, or you may even start it a couple days before or even during the new pack of pills. This can seem strange but is no cause for alarm.
Basically your own hormonal cycle is interacting with the hormones from the pill and it can take a little bit for everything to harmonize, if you will, but don't worry--if you are taking as directed (and it sounds like you are!) you are protected from pregnancy even during this "harmonizing" period. Like horriblegoose said, just start your new pack on time and proceed as usual.
Typically your cycle will normalize within 6 months of starting a new form of hormonal birth control. By then you will start to get a sense of what's usual for you on this pill in terms of when you start your withdrawal bleed, how heavy it is, etc. (For example, I'm on an extended-cycle pill and I only have a withdrawal bleed every 15 weeks, and it's VERY light and starts very late. This didn't become clear until I had gone through a few 15-week cycles).