Even though 'techinally' you started your period yesterday, you could still start taking your pills today. It really won't make a difference. The reason why your gyno told you to start your pills on the first day of your period is because if you start your pills on the first day of your period you're fully protected on birth control. If you choose to start on a random day (a day when you're not on your period) you have to use a back up method for a couple of weeks before birth control takes effect.
Starting your pills today really won't make much of a difference. I'd say go ahead and start them. When i first started taking birth control i didn't start my pills on the day of my period like my gyno said. I just started taking them at a random time, whenever i felt best. It didn't cause me any problems.
As for your vacation, every woman reacts different to birth control. The begining of your fourth week of pills SHOULD be your placebo week, and you'll begin to have your period that week. It might be ligher than usual because you were on your period at the begining of your pills too. It shouldn't effect your vacation too much though. good luck