I am considering starting a short animation cartoon for the web and I
am trying to determine which route to go.  Would 3D via Maya or 2D via
flash be the fastest route. Or is there another program that I should
be considering.   If I go 2D would it become outdated soon
since everything seems to be going 3d?  Does anyone have a sense for
where online animation is going?

2d is not going out of style anytime soon. There is also many ways to
make 3D look 2d with use of cell shaders etc(which is good for flash
as this cell style animation can be rendered to vector information)
The medium doesn't matter as much as the storyline and aesthetic sense
of the artist. 3D is a serious challenge in terms of learning curve
and you have to learn many different departments in order even begin
doing something like animation(modelling, animation, texturing,
rendering, lighting) you need a fairly hefty machine with a
specialized video card if you are going to be doing any kinda of
serious work.

If you go 2d things are much simpler. You can use flash(although it's
timeline animation tools are fairly weak) use simple animation. And
use photoshop and another vector package like coreldraw or freehand
for creating content. You also don't need a beast of a machine.

But I need to emphasise the the single most important factor is
content, nobody cares if the graphics are awesome and the storyline
not engaging. On the other hand you can find stick figures highly
amusing given the right dialog.

It will depend on wheather you know Maya or Flash because these are
both programs that have steep learning curves. Neither are going out
of style so it is up to you. There is another program you can use that
is called Toon Boom and there is a lot of animation studios that use
this program. And also it would not become outdated as long as you
were to draw in a newer style such as anime or manga, which is cell
shading. Online animation is going towards the 2D flash style. This
can be accomplished with toon boom also.