Basic Question about Newton's Method for Optimization

This is a very basic question about Newton's method for optimization, but I cannot seem to find the answer in any of my searches. If we are using Newton's method (or gradient descent), how do we find a maximum instead of a minimum? Do we just change the sign of the step size to positive instead of negative?

Newton's method for unconstrained optimization finds local extrema. Given a function $f(\mathbf{x}):\mathbb{R}^n \to \mathbb{R}$ which we wish to minimize, Newton's method works by finding a root of $\nabla f(\mathbf{x})$. Absent any other information about the behavior of $f$, if Newton's method converges for some arbitrary initial guess, we don't know whether we've found a local minimum or maximum (or saddle point). Writing the iteration as $$\mathbf{x}_{n+1} = \mathbf{x}_n - H^{-1}\nabla f(\mathbf{x}_n)$$ where $H$ is the Hessian, it sounds to me like you're asking whether we might find a maximum by instead computing $\mathbf{x}_{n+1} = \mathbf{x}_n + H^{-1}\nabla f(\mathbf{x}_n)$, where we have changed the sign of the second term. This will not get us anywhere, as it moves away from a stationary point. To use Newton's method in an unconstrained optimization problem to find a maximum, you would use it exactly the same way as if you sought a minimum, probably with a different initial guess. Wikipedia's entry on Newton's method in unconstrained optimization offers a succinct discussion of the concept.
Descent methods, on the other hand, work differently; gradient descent, for example, updates each iteration by moving along a direction such that $f(\mathbf{x})$ decreases. In this case, it makes sense to change the sign of the update if you are seeking a maximum instead.
Yes, that's exactly what you do. You can think of this sign change as causing you to perform gradient ascent instead of gradient descent (in the case of using a gradient method). Alternatively, you can think of flipping the sign in a gradient method as performing gradient descent in $-f$. By finding a minimum of $-f$ you find a maximum of $f$.