I've written a couple of root finders which employ Newton's method.
My experience is that if you have no clue where a root is, Newton's method will turn on you. Much of my time writing these methods is spent scouring the literature for asymptotics which bracket the roots. Even with the asymptotics, it's not enough to just blindly apply Newton's methods to (say) the average of the brackets; first a few iterations of bisection is required to get the root to an accuracy of (say) 1 part in 100.
As to the second concern about the cost of evaluation of $f$ and $f'$: It is not generally that case that you must evaluate the function and its derivative independently. Generally you wish to make a routine that evaluates both at the same time. This is particularly important for evaluation of (say) a power series where the coefficients must be transferred from cache (or worse RAM) to registers. For a power series, it's easy to write a routine that will evaluate both $f$ and $f'$ at once with very little overhead relative to evaluation of $f$.
Finally, there is a very good reason to switch from bisection to Newton's method, even if you aren't interested in speed. Bisection must evaluate a function very near a root to be accurate. But the condition number of function evaluation is unbounded at the root, leading to large error. Newton's method suffers from this problem as well, but from my observation Newton's method exploits the differentiable structure of the function and will recover every digit correct up to the precision of the type. (I cannot prove this statement but I do have unit tests which show it's validity for a few cases.)