My character bought a Fire spell from the Mage's Guild in Chorrol, it qualifies like "20 fire damage at target". When being applied, it shows all visible signs of functioning, but it causes no damage. Tried it on various targets. Is this a bug?

What level are you?, and which targets have you tried, what armour are you wearing, also does the health bar show up at all on the targeted npc's?

As the more armour you wear the less power your spell is likely to have, check the bottom of the spell menu, it should show a number with a % to it.
Some enemies are resistant to fire.
Lastly 20pts is not that much, especially for a higher level character, so you may only be doing a small amount to their total health.

Destruction spells only work on living objects . If you use them on trees, buildings etc. they do no damage . Try it on a crab or rat . It should kill them instantly . Its just as well they don't work on trees or wooden objects or you would burn up Cyrodill .