When I declare my attackers, that causes the attacker to be tapped. What happens if the attacker I tap has an ability that is triggered by tapping?
Would I have to tap the creature before the combat phase to use the ability (thus making it unable to attack due to it being tapped) or does tapping the creature to declare it as an attacker cause it to trigger its ability?
If you can attack and use the ability depends on exactly what kind of ability it is.
If it is a triggered ability like Fallowsage that says "Whenever Fallowsage becomes tapped, you may draw a card." you get to use the ability anytime that card becomes tapped, attacking is the most common way but any other way of tapping it would also work.
If it is an activated ability that has tapping as a cost like Elvish Mystic that says "{T}: Add {G}." you have to choose whether you are tapping it to attack or if you are tapping it to pay for its ability. This is because tapping the creature is a cost and if it is already tapped from attacking you can't tap it again to pay for the cost.
You can either attack or activate the ability.
What happens if the attacker I tap has an ability that is triggered by tapping?
Such an ability would be worded as follows:
Whenever this creature becomes tapped, ...
If a creature had such an ability (e.g. Fallowsage), it would trigger if the creature becomes tapped for any reason, including attacking. In fact, it's effect would only happen if you have some external means of tapping it (e.g. by attacking).
The thing is, very few creatures have such an ability. You're probably not asking about a triggered ability; you're probably asking about an activated ability. Activated abilities have the form "[cost]: [effect]". (Note the ":" and the lack of a leading "when", "whenever" or "at".)
Specifically, I presume are talking about an ability with a tap symbol in its cost. e.g. Golden Hind or Dakra Mystic.
Tapping does not actually initiate the attack or cause activated abilities to activate. Choosing to attack or to activate the ability is the first thing that happens. Tapping is then performed as part of that process.
Tapping as part of activating an ability is done to pay a cost. Tapping as part of attacking is similar. Like costs in real life, taking out a dollar out of your wallet doesn't allow you to buy two one dollar items.
