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Since we already use gravity assist to launch satellites through the solar system, is there any way we can use gravity assist to speed up a manned trip to a planet such as Titan or Europa, while also avoiding the problem of needing lots and lots of fuel to lift fuel?
What would be a rough time frame for a trip using this method?
In general, Gravity Assists do not reduce the amount of time, unless you are going really far out there. For instance, Galileo took 6 years to make it to Jupiter after making 3 flybys, one of Venus, two of Earth. Comparatively, New Horizons made it there in just over a year, Voyager 18 months, and Pioneer about 2 years. However, there is considerable fuel savings from doing the flybys.
However, if you are already going to pass through the orbit of an object, then a flyby could speed you along. Thus, Voyagers are going faster through space than New Horizons will ever be, because of the extra flyby of Saturn the Voyagers received.
Getting to Mars would not have any candidates for speeding the journey time to it except perhaps the moon, as Mars is the easiest planet to reach from Earth. A flyby of Jupiter would probably make a mission to Saturn faster, if you line it up just right.
A gravity assist (or a slingshot) is one of the many compromises in mission design. Instead of going somewhere directly you go somewhere else first and use the momentum of a planetary body to speed up your own movement thus fitting into a Delta-V budget.
So the crux of the problem is that gravity assists take time. For unmanned missions, this is acceptable since avionics and science instruments are shielded, and can be shut off when not needed.
For manned missions, existence of unforgiving radiation environment due to solar wind and galactic cosmic rays means you have to make the trip as short as possible. Muscular deconditioning and bone loss are also threats that arise in microgravity. What's more, if the trip is shorter, you don't need as much mass dedicated to life support (oxygen, food, clothes, toilet tissue). Thus, in general, gravity assists aren't suited for manned flights.
There are exceptions:
slingshots around giant planets (and their heavier moons) when you want to study the moons,
free return trajectories (in case your propulsion malfunctions) and
Buzz Aldrin's cycler orbit options (with or without powered gravity assists - these are the orbits that revisit say Mars and Earth without much expenditure of fuel) - a bit of a variation on "free return".
On a purely personal note (and if you ask me), I'd prefer a brachistochrone constant 1g acceleration/deceleration trajectory to any gimmick to solve the artificial gravity and rad protection problems simultaneously.
The options for good gravity assists are limited. The most one can gain is the relative velocity of the object one passes by. The Moon's orbital velocity is only 1 km/s. Objects with low mass are less efficient to use too. Some probes staying close to Earth, like the STEREO probes, used the Moon for gravity assist. Dawn and Rosetta used Mars. But Jupiter is perfect with 13 km/s in orbit. All six probes which went beyond Jupiter used it for gravity assist (not counting Rosetta which temporarily is a bit beyond Jupiter's orbit). And one might get an extra push from Io which orbits Jupiter at 17 km/s. But it would of course only be helpful for the second half of a trip to Saturn 10 AU away. However, manned travel that far is very futuristic. Today there exists no plans at all to ever send humans beyond the Moon. Several necessary technologies like space nuclear power and centrifugal "gravity" remain completely unresearched and off the agenda.
