Do I need to complete training to play online?

From AAManual

No. If you wish, you can start playing immediately. However, you will find that your experience in the game will be greatly enhanced as you work through the Basic Combat Training (BCT) and Advanced Individual Training (AIT) training missions. Every training level you complete unlocks a new weapon, equipment item, or skill (e.g. Combat Lifesaver (CLS)) that you can use in-game.