Here's a sort-of-realistic answer:
In theory, the technology to accomplish a brief flyby of Alpha Centauri likely exists now.
A small spacecraft, capable of carrying 10 astronauts and enough provisions and equipment to last the crew for a month or two and nothing more. This craft would be launched from Earth opposite the direction of the Earth's orbit around the Sun - plunging the craft into an orbit just above the Sun. From there, the craft will jettison a nuclear device that will explode just behind it - propelling the spacecraft into escape velocity from the Solar System. The low altitude maneuver above the Sun is meant to milk their their high orbital velocity for all it's worth and take advantage of the Oberth Effect.
Several more of these nuclear propellers will be dropped on the trip out of the Solar System. Enough to bring the ship to a comfortable cruising speed so that it can safely negotiate the Oort Cloud. Once clear of the Oort debris, more of those nuclear propellers can be dropped, allowing the the craft to build up relativistic speed - averaging about 25% to 30% of the speed of light. At that speed, our craft travels the 4.4 light years to Alpha Centauri in 14-18 years. In order to maximize speed, mass must be minimized. Any waste (trash, dehydrated fecal matter, etc) must be immediately jettisoned to cut down on the weight.
Meanwhile, the crew is sound asleep in some form of cryostasis to keep the crew from aging or consuming precious food rations, possibly using a protein found in icefish blood to keep their cells from bursting during freezing. Onboard computers handle everything during the trip, except when one of the crew is awakened to inspect the vessel and ensure the mission is continuing as intended.
Upon arrival in the Alpha Centauri System, the computers awaken the crew 10-12 days prior to their closest pass above the star. The crew spends every waking minute conducting as much scientific observation and experimentation before they reach the closest point to Alpha Centauri. If there are any planets - they might try to spare some propellant to fly past one if possible. All decisions will have to be made onboard without consultation from because it will take almost nine years to get any response back from Earth. Because of this, any decision to modify the flight plan - or any aspect of the mission for that matter - will need to be taken with extreme caution.
As the craft slingshots over the 'surface' of Alpha Centauri, the craft drops another nuke as was done above the Sun - building up even more speed for the return trip back to Earth. The return trip averages a speed of 30%-35% of the speed of light. The astronauts go back into stasis as the craft hurdles back to the Sun for the next 13 years.
Once the craft returns through the Oort Cloud into the Solar System, the crew must decelerate before it reaches Earth, otherwise they face a collision with Earth that would destroy entire countries. This might be done with a series of flyby maneuvers using the gravity of Jupiter or Saturn to bleed off excess velocity, followed by very gentle aerobraking maneuvers through the upper atmosphere of the gas giants. This process could take years depending upon the position of the planets around the Sun.
Eventually, the spacecraft would slow down enough for the craft to safely return to Earth. After 40 years of space travel( plus an additional 1-2 years due to the effects of time dilation at relativistic speeds) the astronauts return home with little more than observational reports and some cool film footage/photographs.
The end.