As a student pilot with 100 hours and 300 landings, not to mention interest in engineering and physics, I love questions like this.
My first recommendation is get a copy of Stick and Rudder.
It's a delightful book, used by pilots for over half a century to understand the basics of flight.
My second recommendation is take an introductory flying lesson. It costs around $200 and is the most fun per buck you can have honestly.
Now to try to answer.
(Almost any question like this can be understood with a simple balsa wood glider or a paper airplane.)
First to understand is that in straight flight at constant speed, for a normal airplane with single main wing and a tail, the wing pushes up and the tail pushes down.
The wing supports not only the downward weight of the airplane but also the downward force on the tail.
The plane is nose heavy, and if the tail were suddenly chopped off, the plane would immediately go down.
Since the tail is some distance behind the wing, its downward lift creates a moment that holds the nose up.
The lifts, up and down, and thus the moment, are proportional to speed squared.
If for some reason the plane slows down, the moment decreases, the nose drops, and the plane speeds up.
If it speeds up, the moment increases, the nose rises, and the plane slows down.
This is the first thing you need to understand about airplanes - how the nose-heaviness and the downward lift on the tail cause its speed to be stable.
The exact speed it seeks depends on the angle of the tail surfaces.
The more they are canted upward, the slower the plane flies, and downward means faster.
(In case you're wondering how the wing gets enough lift when it is flying slower, it's because the whole plane tilts up, giving it more angle of attack.)
I know you are asking about turning, but this is the first thing you need to understand.
Second, how do you make it climb or descend?
That's what the throttle is for.
Just like in a car, you need more power to go up a hill, and less to go down.
In a plane, if you increase power, it speeds up, raises the nose, settles back to its natural speed, and keeps climbing.
If you decrease power, it slows down, drops the nose, and then settles to its natural speed, but on a descending slope.
OK, you were asking about turns.
The way you turn a plane is by putting it into a bank.
(With the ailerons, by monentarily turning the yoke left, and then re-centering it.)
Suppose you bank it 30 degrees to the left.
That tilts your lift vector, so that half of it is accelerating you to the left, and .866 of it is pushing up against gravity.
To keep gravity from winning, you apply back pressure on the yoke, creating more lift.
A turn has much in common with a climb, unless you want to descend at the same time.
You also need to give it some more power, to maintain speed through the turn.
Here's a 60-degree bank: where you have to pull 2Gs of lift in order to maintain constant altitude, and add quite a bit of power. That's a pretty stressful maneuver, and you can't do it at slow speed because if you pull that hard it will stall. That shows you the similarity between a turn and a climb.
You were asking why doesn't the plane just accelerate sideways rather than turn?
If it did, then pretty soon it would be feeling this sideways wind on it.
Every airplane is a weather-vane. It turns into the wind it feels.
So, roundabout, that's the answer to your question.