Combine solar sails with planetary slingshotting and ion thrusters
You can't use a solar sail to accelerate towards the Sun. At best, you can slant the sail to accelerate at nearly a right angle to the sunlight, basically increasing or decreasing orbital speed (which will also work to increase or decrease the orbital radius).
Hence, you could increase orbital speed this way when moving from the inner planet to the outer planet and decrease orbital speed when moving in. You could feasibly adjust the acceleration to adjust for the planet's varying positions along their orbits (sometimes, they will be on the same side of the Sun, sometimes on opposite sides, and anywhere in between).
When arriving at the outer planet, you can reduce your velocity with a close pass to the planet (slingshotting), which theoretically can reduce velocity by up to twice the planet's orbital velocity, though lower in practice if you want to avoid catastrophic atmospheric friction and tidal strain. Disembark after this passage to reduce the amount of deceleration your shuttle has to do. When arriving back at the inner planet, use a similar maneuver to increase orbital speed by up to twice the orbital velocity of that planet. Here, disembark before this maneuver, since your orbital velocity is likely greater than that of the planet.
There are some constraints to this method. The fastest way to move inwards is to totally kill your orbital velocity, basically free-falling inwards. This is limited by the gravitational pull of the Sun at the distance you are, which is likely to be a very small fraction of one gravity. At Earth's orbit, the Sun's gravitational pull is 0.006 m/s/s (meters per second squared), and this pull decreases as the square of the distance from the Sun. Moving from Mars to Earth, the average pull will be around 0.0036 m/s/s. Moving from the (average) orbit of Mars to that of Earth will then take a minimum of 76 days. Going in from Saturn to Jupiter will take far longer; on the order of 3.5 years.
There are no similar constraints going outwards. Giving a large enough solar sail, your acceleration is only limited by the mass per square meter of the sail. Given a light (and strong) enough material (unobtanium), you could feasibly maintain an acceleration of 1 g. Accelerating at 1g all the way, you could move out from the orbit of Jupiter to the orbit of Saturn in just 4-5 days! (arriving at very high speeds, though). Or you could if there was no solar wind. The solar wind, consisting of charged particles emanating from the Sun, has a much higher energy density than sunlight and will begin to brake your solar sail once you reach outbound velocities exceeding around 400 km/s (the average speed of main component of the solar wind). This increases the minimum travel time out from Jupiter to Saturn to around 20 days. Your ship will need to decelerate from 380 km/s (400 km/s minus twice Saturn's orbital velocity of 10 km/s).
In other words, moving out can be done in a matter of days, but going in will take months or years, depending on how far out you travel from. To make your system feasible, you probably need some kind of engine to provide thrust going in (with solar sails furled or slanted perpendicular to sunlight and solar wind). A fusion-powered ion thruster is probably your best bet, but your ship would have to tank up on He3 before going in. Such a thruster can also decelerate your spaceship going out, arriving at your destination at near-zero speed.
To sum up: A solar sail will help you accelerate going and to adjust the orbital speed of your solar spaceship to match that of the planet you arrive at. However, you will need a thruster (ion or otherwise) to kill your final speed going out and to give you speed going in (where braking can be handled by solar sails). The good news is that you don't have to worry about matching the speed of your payload / passenger module to your solar spaceship, since it will have to kill its speed anyway before reversing direction.