Building a gas-giant colony ship is not as difficult as it looks.

1. Build a fusion candle. It's called a "candle" because you're going to burn it at both ends. The center section houses a set of intakes that slurp up gas giant atmosphere and funnel it to the fusion reactors at each end.
2. Shove one end deep down inside the gas giant, and light it up. It keeps the candle aloft, hovering on a pillar of flame.
3. Light up the other end, which now spits thrusting fire to the sky.
4. Steer with small lateral thrusters that move the candle from one place to another on the gas giant. Steer very carefully, and signal your turns well in advance. This is a big vehicle.
5. Balance your thrusting ends with exactness. You don't want to crash your candle into the core of the giant, or send it careening off into a burningly elliptical orbit.
6. When the giant leaves your system, it will take its moons with it. This is gravity working for you. Put your colonists on the moons.

For safety's sake, the moons should orbit perpendicular to the direction of travel. Otherwise your candle burns them up. They should also rotate in the same plane, with one pole always illuminated by your candle (think "portable sunlight"), and the other pole absorbing the impact of whatever interstellar debris you should hit (think "don't build houses on this side")

Whether or not your gas giant heats up to the point that it ignites and turns into a small star depends largely on how much acceleration you're trying to get out of your candle. Remember, slow and steady wins the race!

Addendum to Note: Larry Niven suggested that such an arrangement could be used to move rocky worlds from one orbit to another, and he wrote a novel entitled A World Out of Time in which the Earth was moved with the help of giant candle they'd shoved up Uranus. I'm not making this up.