At first, it's exciting, terrifying, wonderful, stressful, and better than you could have ever dreamed.
It's much the same as driving a car, welding, or really any technical skill.
You will learn how to do everything; muscle memory will be ingrained.
Right now, you have to actively think very hard about -everything- you need to do, and probably even worry about stuff you don't have to do.
As you gain experience and confidence, things will become second nature.
You'll start a turn, and realize that your feet automatically input the correct amount of rudder without even thinking about it.
Other things you will learn techniques for, Like you were saying with the climbout, eventually you'll learn how your aircraft responds, -that is a very big part of the learning curve-, and eventually, a few flicks of trim will perfectly maintain your speed, and leave you free to correct your course for wind.
I just went up with a friend today, their first time flying. We went on a cross country. I gave them some control during enroute, and at first they were over-correcting for turbulence, and quite jerky with inputs. After about 20 minutes of flying however, they were able to keep it +/- 10° 200' and smooth and level when the wind picked up.
After we landed, they commented that at first, they were freaking out; They were flying a plane! But in time, they learned how it responded, how little actual control inputs you need, and were more confident and less scared.
With time the sensory-overload will lessen. However, you don't want to become complacent.
You will still think about what you're doing. but you will know what to expect, and be able to 'think ahead of the airplane' to anticipate maneuvers, requirements, and tasks.
That's why you train; to learn how to do everything. And that's what your instructor is there for. Ask him anything.
And one last thing;
Welcome to the club. :)