I'm really surprised if this question isn't a duplicate, but since I failed to find one that asked about the Fibonacci sequence rather than someone using it as an example, I'll answer.

The most natural approach, besides using the built-in [`Fibonacci`](http://reference.wolfram.com/mathematica/ref/Fibonacci.html) function, recursion:

    f[0] = 0; f[1] = 1;
    f[n_] := f[n] = f[n - 1] + f[n - 2]  (* note memoization *)

    Array[f, 10]

>     {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Better performing may be [`Nest`](http://reference.wolfram.com/mathematica/ref/Nest.html) and [`NestList`](http://reference.wolfram.com/mathematica/ref/NestList.html):

    fibonacciList[n_] := Module[{x = 0}, NestList[x + (x = #) &, 1, n - 1]]

    fibonacciList[10]

>     {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Another useful way uses [`LinearRecurrence`](http://reference.wolfram.com/mathematica/ref/LinearRecurrence.html):

    LinearRecurrence[{1, 1}, {1, 1}, 10]

>     {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}

Hopefully these examples inspire you.

----------

I now note that you request the sequence starting from zero.  Most of these are easy to adapt or modify.  The first one is simply:

    Array[f, 10, 0]

>     {0, 1, 1, 2, 3, 5, 8, 13, 21, 34}

For the second you may instead write:

    fibonacciList2[n_] := Module[{x = 1}, NestList[x + (x = #) &, 0, n - 1]]

    fibonacciList2[10]

>     {0, 1, 1, 2, 3, 5, 8, 13, 21, 34}

The last one merely needs the proper seed:

    LinearRecurrence[{1, 1}, {0, 1}, 10]

>     {0, 1, 1, 2, 3, 5, 8, 13, 21, 34}

----------

Finally, taking the question at face value you can modify your code to return `fPrev` rather than `fNext` to start from zero:

    fibonacciSequence[n_] := 
     Module[{fPrev = 0, fNext = 1, i = 0}, 
      While[i++ < n, {fPrev, fNext} = {fNext, fPrev + fNext}];
      fPrev
     ]
    
    Array[fibonacciSequence, 10, 0]

>     {0, 1, 1, 2, 3, 5, 8, 13, 21, 34}


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Addendum for rcollyer:

    $fibList = {0, 1};
    fibonacciList[n_] /; n <= Length@$fibList := Take[$fibList, n]
    fibonacciList[n_] := $fibList =
      $fibList ~Join~ 
       Module[{x = $fibList[[-2]]}, 
        Rest@NestList[x + (x = #) &, $fibList[[-1]], n - Length@$fibList]]