Humans can destroy the planet, but they can’t destroy the universe, because the universe includes everything, including the humans themselves! They’d have to destroy themself first in order to destroy the universe =)
The main problem of getting to the end of the universe is how to travel there. Light speed travel (or warp speed like in Star Trek) are the most popular methods, but I discussed some of the problems with them here.
The other issue is that the universe is still expanding from the big bang. The edge of the universe is racing away from us at 74.2 ±3.6 kilometers/second/megaparsec. That means for every megaparsec (about 3 million light years) you go out, the Universe is expanding 74.2 km/sec faster. It makes it harder and harder every second to ever foresee us travelling to the edge. Some theories also say that the universe doesn’t have an edge, and that it is like we are standing on the inside of an expanding balloon. If you headed off in a straight line in one direction you’d eventually end up back where you started! Having said that, it is possible that the universe will begin to contract at some point in the future, bringing the end of the universe back towards us. The chances of humans still being around at that point are quite slim!