Gareth Southgate's side secured their spot at the tournament with a 4-2 aggregate victory over Croatia earlier this week, but Moore is certain that there is more to come from the team in the Czech Republic.
"In the dressing room, everyone's happy but we're not too over the top. This is the start of an achievement that we want to get to," the Leicester City defender, who scored in the second leg against Croatia, told reporters.
"Qualifying is great, but we want to do much better than that, we want to go far in the tournament and show what we're all about.
"We've set out what we aimed to do, which was obviously to get to the Euros so that was job done and we put in a good performance as well."
England have failed to get out of the group stage in the previous two tournaments.