There is no kind of spacecraft that we can envision building based on currently known science and engineering that would be able to reach another galactic cluster in anything resembling a reasonable period of time. Even with highly optimistic assumptions, even the nearest globular clusters are out of practical reach.
Consider that the Milky Way is about 75,000 lightyears across, and that we are approximately 25,000 lightyears from the galactic center:
On the practicality of FTL travel
It's important to note that the current best idea for faster-than-light travel, the Alcubierre drive, is a purely hypothetical construct based on one solution to Einstein's equations; basically, it's the answer to "what happens if we plug in matter with negative mass?". We have no idea how to make matter with negative mass, and the whole concept is purely mathematical; it could be everything from perfectly viable and practical once we figure out the engineering problems, to Einstein being wrong, to this being a case that Einstein's equations simply don't account for like how Einstein's formula for graviational time dilation breaks down close enough to large masses (a phenomenon known as the Schwarzschild radius).
Add to the above that every form of faster-than-light travel or communication at the very least introduces causality paradoxes, referred to as the tachyonic antitelephone thought experiment (source citations in the linked Wikipedia article):
A tachyonic antitelephone is a hypothetical device in theoretical physics that could be used to send signals into one's own past. Albert Einstein in 1907 presented a thought experiment of how faster-than-light signals can lead to a paradox of causality, which was described by Einstein and Arnold Sommerfeld in 1910 as a means "to telegraph into the past".
This of course isn't affected by whether you restrict yourself to FTL travel but not communications or allow for both, because a FTL craft can always carry a physical message, enabling communications.
Assuming you meant globular clusters
Based on a conservative estimate, assuming particularly that the second image above is representative, the nearest globular cluster might be something like 2,000 lightyears away from us. Based on Wikipedia's introduction on the subject, this is probably highly optimistic, which only makes this harder.
If we could find a way to make it to 95% of the speed of light, which is a very Hard Problem ™ but within the realms of science, a 2,000 lightyear trip will take 2,100 years (not accounting for time dilation, which starts to become significant at these velocities). If we launch in 2100, we will be receiving the first data back some time around year 6200 (because the ship arrived in year 4200 and a speed-of-light transmission took another 2000 years to get back to us), and the spacecraft will return at the earliest around year 6300.
If we could find a way to make it to 100 times the speed of light, which is still solidly out of the realms of science let alone engineering (and also causes all kinds of headaches like possible violations of causality), we could make the trip to the nearest globular cluster in about 20 years and would receive data no earlier than another 20 years later when the ship (or a data pod) returns. If we launch in 2100, we will have data by year 2140 or so (and we will detect the ship in the globular cluster some time around year 4100; hopefully we will have good records of the mission, lest we might take that as signs of extraterrestrial life).
Assuming you meant galactic clusters (as you say to be the case)
This is far, far worse than the case of globular clusters.
The Milky Way, Andromeda and a number of smaller galaxies (for a total of at least 54 galaxies) all belong to what is known as the Local Group. The Local Group spans about 10 million lightyears. When considering intergalactic travel, this is the least we would have to concern ourselves with. The Local Group, in turn, forms a part of the Virgo Cluster.
The Andromeda galaxy, which is our closest intergalactic neighbor, is about 2.5 million lightyears away. If we can figure out a way to reach and sustain 100 times the speed of light (again, well outside of science and engineering at present), then the trip to Andromeda would take 25,000 years. A return craft could then get data back to Earth in the same timeframe, so if we launch in 2100, we get data back by year 52100 or so.
For comparison, humans left Africa 60,000 years ago and human agriculture only developed about 10,000 years ago. We can't even imagine what human culture will be like in 50,000 years.
Reaching nearby stars (say, up to 50 lightyears away or so) is reasonable even with sublight speed spacecraft. Such a mission would take up to a few centuries at a somewhat high fraction of the speed of light, but is within the realms of theory, if perhaps not practice at present.
With extreme advances in physics and engineering (including figuring out faster-than-light travel), based on highly optimistic assumptions, reaching a nearby globular cluster in something resembling a reasonable amount of time might possibly be realistic. But as I said above, we have no idea how to do faster-than-light travel; the best we have is a set of solutions to a set of equations that might not even apply.
Reaching a different galaxy, let alone a different galactic cluster, is not realistic at all unless we have spacecraft that can travel at many orders of magnitude lightspeed. Even 1,000 times the speed of light makes such a trip impractical; we'd have to get to 10,000 times the speed of light or so for it to be in any way realistic, at which point a trip to Andromeda and back would take around 500 years. We can't even realistically get to 1/10,000 the speed of light (30 km/s) with current spacecraft, so we've got about 100,000,000 times our current best efforts to go.