Question

f(x) = 6x^2 -8x +3

how would I find the most general antiderivative of the function. I also have to check my answer by differentiation.

Would I have to find the derivative first? (I got 12x -8)
What are the steps to solving a problem like this?

No, don't do the derivative first. Do that on the integral, to check your answer.

I use the word "integral" for what you are calling "antiderivative". You can say I am old-fashioned, but I don't see a need for a longer word.

The general rule for integrating any
a x^n term
is that the integral is
a x^(n+1)/(n+1)
When you have a polynomial, as you do here, the integral is the sum of the integrals of the terms.

The first term of the integral is
6 x^3/3 = 2 x^3. Take the derivative of that and you get the original 6 x^2 back, so it is the right answer.

Now you do the other terms, and add them up.

Remember that the most general integral has an arbitrary constant term added.