Theoretically, you just need to keep records and receipts as the mfg cannot require you to use their dealers to keep the warranty in effect. Since it's a new car, start a maintenance logbook - bookstores sell them or you can find templates online. Or go with fully online records like fuelly.com. The logbook entry should certainly document the date, car odometer reading, and a good description of the service performed, including things like what viscosity and type of oil was used. Attach any receipts for fluids/parts to the logbook entry.
While DIY is certainly Okay, I do think there is some merit to using the dealer for service during the warranty period so there is no excuse for Mazda and/or the dealer to not warranty something using the argument that a service was not properly performed. It just removes an opportunity for argument. And if you need the dealer to "go to bat" for you with Mazda, they're more likely to do so if they've serviced the car. It's a tradeoff between expense and potential hassles.
If you want to do more-often oil changes but also want to have dealer service records, one option is to halve the normal interval, do the extra oil changes yourself (using an OEM filter), and use the dealer for the others. Keep the extra oil changes "secret".
- Mark