There are only a few basic rules you need to know. The most important rule being one of Magic's Golden Rules, APNAP order rule.
101.4. If multiple players would make choices and/or take actions at the same time, the active player (the player whose turn it is) makes any choices required, then the next player in turn order (usually the player seated to the active player's left) makes any choices required, followed by the remaining nonactive players in turn order. Then the actions happen simultaneously. This rule is often referred to as the "Active Player, Nonactive Player (APNAP) order" rule.
The other basic rules is covered in Timing and Priority. The Active Player receives priority at the beginning of phases or steps, and after a spell or ability resolves from the stack. Additionally, players do not give up priority until they choose to pass.(i.e. they can chain as many spells/abilities in a row as they wish before relinquishing priority)
116.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don't get priority during the cleanup step (see rule 514.3).
116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
116.3c If a player has priority when he or she casts a spell, activates an ability, or takes a special action, that player receives priority afterward.
The last priority issue is covered under Handling Triggered Abilities. Since triggered abilities can trigger anytime, a rule is needed for how to assign priority and order triggered abilities on the stack. Just before a player would receive priority, any triggered abilities that are waiting to be put on the stack are placed on the stack in APNAP order (the AP's triggered abilities will resolve last, since they are on the bottom of the stack).
603.3b If multiple abilities have triggered since the last time a player received priority, each player, in APNAP order, puts triggered abilities he or she controls on the stack in any order he or she chooses. (See rule 101.4.) Then the game once again checks forand resolves state-based actions until none are performed, then abilities that triggered during this process go on the stack. This process repeats until no new state-based actions are performed and no abilities trigger. Then the appropriate player gets priority.
The only remaining types (besides the Loyalty Ability you already mentioned) of are Turn-Based.actions, State-Based actions, and Spell Abilities. No player receives priority while handling these.