Mild:
* Irritated
* Frustrated
* Disgruntled
* Peeved
* Cross
* Aggravated
* Exasperated
* Displeased
Moderate:
* Upset
* Bothered
* Vexed
* Chagrined
* Indignant
Strong:
* Furious
* Enraged
* Infuriated
* Livid
* Outraged
Other related words:
* Disappointed
* Discouraged
* Disheartened
* Dismayed
* Troubled
The best synonym for "annoyed" depends on the specific context and the level of annoyance you want to express.