着る, 履く, and 被る: choose a verb by body area
Japanese divides actions covered by English wear according to where an item is worn. Also distinguish putting an item on from the resulting state expressed with ている.
Upper body, lower body, or head
着る is used mainly for garments covering the upper body or the whole body, such as shirts, coats, and dresses. 寒いのでコートを着ます。 describes putting on a coat because it is cold.
履く is used for trousers and skirts worn below the waist, as well as shoes and socks. Both ズボンを履く and 靴を履く are natural. For a hat worn on the head, use 被る, as in 帽子をかぶる.
Use ている for the state of wearing
Use 着ています for the current state of wearing clothes. あの人は青いシャツを着ています。 says that the person is wearing a blue shirt; it does not only describe the instant of putting it on.
The corresponding states are 靴を履いています for wearing shoes and 帽子をかぶっています for wearing a hat. Learn the te-forms 着て, 履いて, and かぶって together with the body area.
Do not use 着る for every wearable item
Even if another language uses one general verb, do not make スカートを着る or 靴を着る your basic pattern. Choose スカートを履く and 靴を履く according to where the item is worn.
Other accessories take still other verbs: 眼鏡をかける for glasses and ネクタイを締める for a necktie. Check the actual item and the resulting state instead of translating every instance of wear as 着る.
Quick check: What completes 今、帽子を（ ）います to mean that someone is wearing a hat?
Show the answer and why
Use かぶって: 帽子をかぶっています。 A hat is worn on the head, so the verb is 被る, not 着る or 履く.