Can we travel deeper in the universe?

Traveling in the universe has been the most common thing and appelaing mostly about it is that you can travel around the universe and alos that you can travel up so much deeply that you must essentially be able to travel  at speed of light and also such speeds that are most common about you and that you must perhaps be liking that since doing that we might be able to do it in our lifetime.
Humans space program became into existecne in the 1960s where American President and the President of t he scientific governent- based agency told that they will bring up Astronauts into the  lunar surface by the end of the decade , the first ever humans to have gone with a spaceship  that carried the human crew to the Moon, was just 312,000 miles away, so how many hours can we travel now into the universe?