From the notice board in the Gran Soren inn you can accept escort quests. These require you to protect an NPC as you travel from the inn to another point on the map. One of the easiest to do is the escort quest for Mercedes Marten. Unlike most other NPCs you can escort, she has a weapon and armour and will actually fight if there is danger. To travel with the least amount of trouble, travel during the day and keep to the roads.
An easy method of obtaining this is to place a portcrystal near the escort's required destination, which you can first obtain after the quest "Griffin's Bane". When you defeat the Griffin, the portcrystal will be near the treasure. You can also purchase these in NG+ for 200,000 G at The Black Cat. So after placing the portcrystal use a ferrystone to get back to Gran Soren, begin the quest, then use another ferrystone to travel to the portcrystal you placed. You will have completely avoided harm and unlocked the achievement.