> It's much better to save fuel and send more stuff than to get there a few months earlier.
Yes. For cargo and probes/rovers. Not for humans. The faster we can arrive, the less supplies are required and the risk of something going wrong while the nearest help is light minutes away is lessened.
> For your actual question...you can get faster routes with more fuel
Not fuel, but more Delta-V. Which may or may not imply on more fuel. Options are: more fuel (rocket equation bites here), less weight, more efficient thrusters, aka higher ISP (see the nuclear propulsion sibling comment).
What this gives you is the ability to use other trajectories besides pure Hohmann Transfers - which are efficient, but slow.
Ideally we would send supplies ahead of time with the most efficient transfers possible, then get humans there as fast as we possibly can. Otherwise this would take a while, around 400 days, one way. So over two years just to get there and back, plus wait time until the next launch window.