Like you, i was driving a 14 year old vios before upgrading to Mazda3, so i may (or may not) relate better to you (as you're also upgrading from a vios). Every driver has different driving styles, and there's no issue with what you're currently practicing.

As reference, so here's what I usually do with a car regarding how the gearbox responds to engine input. I'll memorise the automatic shift points when driving light footed. This way, i'll know what is the minimum speed/RPM range that the gearbox will shift up another gear. Due to efficiency reasons, there's always a hard coded shift point to upshift as soon as possible at certain minimum car speeds. After I have a hang with it, then I know i have to avoid having too light input when I need to keep the current gear at certain speed. Another way is to put the gearbox to manual mode by pushing the gear knob to the right while in "D". It will not upshift until you either put it back to place or you push up or down for your desired gearing.

Some other observation points for you to consider :
- The car has an angle sensor build into it, so it won't upshift if you're at a certain upward incline unless you manually do it with pedal shifter or the do it manually using the gear knob method.
- It will downshift from 6th to 4th gear if you brake quite significantly. To objective is to use a bit of engine breaking to slow the car faster and another is for a more responsive acceleration after that.
- Like it or not, at lower RPM ranges, the gearbox will not be as smooth shifting as the vios does. There's always a slight jerk when it changes up a gear.
- There's a click point on the pedal even after you reach maximum pedal travel. You can feel the click. It engages maximum acceleration potential and will (unfortunately) upshifts just before the redline around 6500k RPM.

One more thing regarding the noisy tyres, go get the pressure checked. Mine came with 320kpa when delivered. In contrast, the maximum manufacturer recommended pressure is 280-300kpa. So bleeding out a bit of air to reach around 250kpa should help a bit. But i put mine to 280kpa even though i'm driving alone most of the time. Higher air pressure will lead to stiffer tyres which will improve handling and lower rolling resistance a bit to save some fuel.