You can accelerate prior to the interstellar portion of the mission. This could be done with booster rockets like many real-world missions, or it could be done with a space tug (which is really just a variation of a booster rocket). You could also get fancy and use some kind of rail-gun setup, with magnets or lasers or whatever mounted to orbiting stations in the departure solar system and each station adds a bit more momentum to the ship.
Whatever method you use, the ship would just drift through space until it reaches its destination. If it's the first ship to arrive, it's probably1 going to have to use one or more rockets to enter planetary orbit and eventually land. Otherwise you don't need to use any major thrust on the way, though, as Michael Kjörling points out, you'd likely want some thrust to make course corrections along the way2. If it's not the first ship to arrive, previous ships could have the same kind of fancy setups to slow the ship down, but those will take a lot of resources to build, and won't likely exist until many ships have arrived and settled into the new star system.
That said, your ship can't have a completely closed setup. EM energy will leak out into space, plus entropy says every time your life cycle loops around, you don't have quite as much extra energy. So you need a power source that will keep the ship warm, feed the plants, etc. for the duration of your trip, although a decent-sized batch of nuclear fuel should suffice depending on the size of the ship and crew.
The other consideration is that you'll get there faster if you keep accelerating to the halfway point. This means you need much more fuel to accelerate, then even more fuel to decelerate, and even more fuel to get all that fuel out of the solar system, and even more fuel to get all of that off the origin planet's surface. It will be a lot more expensive, but less time in transit means less opportunity for failure, and fewer generations living in the ship.
1 I think you could technically use a giant lightsail to airbrake once you get to the other end, but my gut feeling is there's not enough acceleration there to slow you down from the very high speeds you'd want to traverse interstellar space at. However, the wikipedia article mentions a hypothetical way to propel a spacecraft to the inner Oort cloud (about 0.8 ly from Earth) in 30 years, so similar technology might be used to reach Proxima Centauri (about 4.2 ly from Earth) in a reasonable timeframe using no rockets.
2 Tiny errors in velocity at the beginning of the trip could cause you to miss the other star system entirely if not accounted for. As seen in Brendan's answer to another question, even the moon flights made a couple small course corrections along the way, and that's about 100 million times closer than the nearest star system.