Sugars which are Carbohydrates and vice-versa?
You need to make a distinction between polymers and monomers (or dimers).

Typically, sugars are monomers like glucose, maltose, or fructose. Sucrose (table sugar) is a dimer. These are all carbohydrates (made up of carbon, oxygen, and hydrogen).

Polymers of these sugars are also carbohydrates. For example, starch, glycogen, and cellulose.

Cellulose, which you called fibre, tends to be structural, and is only found in plants (think celery, or wood, or paper). Starch, is also synthesized by plants, and could be used for energy storage (as I recall). In animals, monomeric glucose is stored as glycogen in skeletal muscle, where it becomes available as energy for short bursts of energy.