From a traveling point of view, sending a probe to Pluton (New Horizons) took us 10 years and the probe had a nice pull from Jupiter. The closest exoplanet is 6 700 times farer than Pluton.
From a communication point of view, the probe/rover would not be able to send us back any kind of signal. Even if we were able to communicate, a round-trip message would take 8 years.
And remember, we are not always successful when sending a probe/rover to Mars, which is the closest planet to Earth.
Remember, we are already using super sensitive telescopes for looking at the star nearby the exoplanets.
The acceleration itself wouldn't be too bad:
299792458 m/s / 2 = 149896229 m/s
In 10 years that would be: 149896229 m/s / 10 y / 365 d / 24 h / 60 m / 60 s = 0.475 m/s2
To propel a spacecraft, due to the law of conservation of momentum, if you want to go in one direction, you have to push something (the propellant) in the opposite direction.
It's the momentum of the ejected stuff (mass*velocity) that matters for maneuvering in space. You have to carry that propellant with you and accelerate it. You can use specific impulse to compare efficiency of rockets (the higher, the more efficient is the rocket in creating forward thrust with the same amount of propellant). The best engines we have are ion thrusters because we can accelerate a small amount of gaz (not very heavy to carry) very fast => big momentum. But thrust is limited, this why you can not lift off from Earth with an ion thruster.
Nuclear energy is just used to power spacecrafts in electricity when you can't use solar panels.
The probe would have to be self-navigating to a certain extent, because we can't locate and measure the exoplanets with precision and won't be able to communicate in near-real-time to guide it in. It would have to be built for long endurance and have the capability to repair itself, too.
The final practical problem is that we wouldn't know for decades whether the probe had been successful, partially successful, or a total waste of time. It could land on an exoplanet and do wonderful science but not be able to communicate due to a broken radio. Or it could blow up in year one and we might not notice.
I suspect the first insterstellar probes instead will be inexpensive and disposable "fire and forget it" types launched with a booster rocket (maybe like the one Elon Musk is building) and no intention of decelerating or landing. They might spend centuries in flight and be designed to send pictures and data back to whoever might be listening in the future. If they're cheap enough and we have extra thrust, I could see rich people and governments launching these in hopes of a sort of future glory, knowing that each individual probe faces long odds but a thousand of them should produce at least a few winners.