DIAGNOSIS Yeah, this one's common. The printer is doing a lengthy self-cleaning and priming cycle at the start of every job, especially if it hasn't printed in a while. It's trying to prevent the clogs we talked about earlier.
FIX You can't fully stop it, but you can minimize it. In your computer's printer preferences, look for a 'Quiet Mode' or 'Eco Mode' and disable it. These modes slow down the printer even more. Also, in the same menu, set the 'Default Paper Source' to 'Auto' if it's set to 'Rear Tray', as the printer sometimes checks both.
IF THAT DIDN'T WORK Make sure you're printing a simple document to test. Complex graphics or documents from web browsers force the printer to process data longer before it starts physically printing.
HEADS UP The first-page-out time is just a trait of this type of inkjet. If you need speed for single-page documents, you're better off leaving the printer on all the time so it stays primed and ready.