In épée, how do I attack without just getting parried or counter-hit?

You cannot make an épée attack completely safe, but you can make it much harder to answer: attack from closer, attack at a moment when your opponent cannot react well, and control their blade or line before you finish. In épée there is no right of way. Whoever hits first scores, and if both hit within about 40 to 50 milliseconds, both get a touch. So any attack that leaves their point in front of you tends to end in a double, and a long attack from far away gives them time to parry or step back and hit.

Start with distance and timing, because most failed attacks start too far out. Steal ground with small steps until a short lunge will reach. Attack as they step forward or just as they finish a step, not while they are settled and waiting. Extend your arm first and keep the point aimed at the target the whole way; a wind-up or a bent arm tells them it is coming. Use the nearest targets too: a quick touch to the wrist, forearm or top of the hand from long distance risks little, and once you have scored there they start protecting the hand, which opens the body.

Then take their blade out of the picture. Attack with opposition: as you hit, close the line with your guard, usually sixte, the outside high line, so their counter-hit is pushed off target. That is what turns a double into a single light. Beat or bind the blade before the final lunge so their point is out of line when you arrive. Against a fencer who always parries, use a feint and disengage: show the attack into the line they guard and go around the parry when they react. Against someone with a reliable parry and riposte, use second intention: attack at a deliberately readable pace to draw the parry, then parry their riposte and hit.

Read the score as well. A double is not always bad: when you are ahead, each one moves you closer to winning. At 4 to 4 in a pool, though, a double no longer counts, so that touch needs a clean single light. Ask your coach for lessons on attacks with opposition and on the feint and disengage, then ask a clubmate to parry the same line every time while you drill going around it. Once it works against a known parry, practise against free reactions.