Realistically this would probably not work too well.
Since you're partially asking for some level of realism, here's the necessary addendum: Dragon-style bodies synergize very poorly with a large body size, especially if you're adding even more limbs to that mix. Your idea is good from an arthropod-like perspective where you have several limbs and can afford to specialize different limbs for different purposes, being a decent Jack of all trades, but flying is a big issue, because your wings need to be strong and your body must ideally be as light as possible. With that in mind, things like extra limbs make this troublesome because instead of having 2 wings with x muscle, you'll have 2 pairs which will most likely have more than x amount, but with added amount of weight from the second pair's skeletal structure. The large amount of limbs you have also means you're less streamlined, which here is twice as problematic since you want them to be both good fliers and good swimmers, and not being streamlined means your swimming speed should take a toll.
In an ideal world where we're still ignoring the square cube law' s annoying rules, but not completely, your centauroid character would probably benefit from a lack of upper body wings and a more wyvern-like lower body, with the wing limbs being the most powerful pair, sharing many traits with some azhdarchid pterosaurs. This would already mean they'd be lighter and more streamlined than if they had the dragon-style 6 limbed lower body with a 4 limbed upper body. The natural weight of the front part, lacking wings, would help balance out the lower body during flight, and they could still benefit from their wings, as the large size likely means they could achieve wings big enough for soaring. Something more or less
(funny enough this is pretty similar to the body shape I gave my own creature, a creature that was much more arthropod-like, but that also needed to be good at burrowing and wasn't a centaur. Taking a look at the fuzzy mole crickets might help you fleshing out your design much like it helped me in mine):
that out of the way, something between a dragonfly and a puffin
Puffins are a very cool type of bird for an important reason: they're good swimmers with a swimming style much like that of penguins, but they're also very good and speedy fliers! Their strategy is their wings, which are adapted for this hybrid locomotion lifestyle, and their habit of beating their wings up to 400 times per minute, or 20 times every 3 seconds (and also, their beaks are used by them as shovels to dig their burrows, making them decent diggers for a bird). Given what we want is something that needs to fly and swim well, this seems like one of the best ways other than modifying the tail and lower limbs for swimming.
At this point I'll say it again however: your character will likely have a major part of its weight located on the lower body, so the pair of wings on the upper body would probably give it a lift it doesn't need, risking imbalancing their flight.
As for however they'd flap their wings, ignoring weight issues, they'd probably be better off doing like dragonflies, which are agile aerial predators who use their 4 wings to their max potential, actually varying in the way they flap their 2 pairs of wings depending on whether they're traveling a long distance, hunting or hovering. Taking a look at their various flapping patters can help you to make the most of your character's 4 winged anatomy for maximum agility.