There are incredible, mindboggling distances involved in merely reaching an exoplanet. Alpha Centauri Bb, the closest known exoplanet, is 4.365 light years away. That's 41,295,000,000,000 km (25,660,000,000,000 mi), or 276,000 AU (the distance from the Earth to the Sun).
Voyager 1 has left most of our solar system behind and is traveling a blistering 17.3 km/s. If it were headed toward Alpha Centauri Bb, it would take nearly 75,700 years to get there.
At interstellar distance scales, the technical challenge is all about attaining speed. Project Icarus is proposing to use a nuclear fusion to attempt interstellar speeds. Beamed propulsion is a proposal to use light sails powered by gigantic lasers.
Another way to get a boost is the Krafft Arnold Ehricke trajectory, recently mentioned in this question, from which I quote:
. . . the best trajectory . . . to get out of Solar System with the highest velocity $V_\infty$ will have to use:
- A gravity assist from Saturn
- A gravity assist from Jupiter
- A perihelion propulsive maneuver (aka firing engines) near the Sun (as close as allowed by the spacecraft's thermal control system).
Even the most optimistic predictions say perhaps within a hundred years we might attain speeds sufficient to get our probes there in hundreds of years. Thus another technical challenge will be designing a probe that will last that long, will still have a functioning power system upon arrival, will still have functioning instruments, radios, etc. This is, of course, complicated by the fact that there is not truly anything like empty space anywhere. The particles in interstellar space are extremely sparse, but when one is traveling at an appreciable fraction of the speed of light, the slightest mote has great damage potential.
Then there's the question of how to decelerate so that when we get to the destination, we don't flash by so quickly that no practical amount of data can be absorbed by our instruments. Most designs call for accelerating to speed, then decelerating. Magnetic sails have been put forth as a possible deceleration technique, but that presupposes sufficient magnetic flux in the interstellar medium to render that feasible.
Getting the data from the exoplanet flyby back to Earth is itself a technical challenge. Assuming your probe manages to capture a scientifically significant cache of data about the target system and planet(s), the vast distance the telemetry has to transmit means a significantly attenuated signal by the time it gets to any receiver in the solar system. One approach might be to send a series of relay probes after the primary vehicle to receive, boost, and then relay the signal.