This is possibly a very simple question, but I couldn't find a definitive answer in the players handbook.
If a character has a speed of 6, can he move 3 squares, perform an action, then move the remaining 3 squares?
No, a standard action cannot bisect a move action.
Each action must be fully resolved before you can begin another action (aside from triggered actions). However, you can use them in any order you like: attack, then move; or move before attacking. There are specific powers and features which provide exceptions to this rule.