Patron Wizard says I have to tap an untapped wizard. That is before the ":", which means that is the cost. However, does there have to be a spell to target to use this ability, or can I just tap an untapped wizard and let the ability fizzle?
You absolutely do have to have a target in order to use the ability.
602.2 To activate an ability is to put it onto the stack and pay its costs, so that it will even tually resolve and have its effect. Only an object’s controller (or its owner, if it doesn’t have a controller) can activate its activated ability unless the object specifically says otherwise. Activating an ability follows the steps listed below, in order. If, at any point during the activation of an ability, a player is unable to comply with any of those steps, the activation is illegal; the game returns to the moment before that ability started to be activated (see rule 717, “Handling Illegal Actions”). Announcements and payments can’t be altered after they’ve been made.
And you have to choose a target in order to cast a spell or activate an ability.
601.2c The player announces his or her choice of an appropriate player, object, or zone for each target the spell requires. A spell may require some targets only if an alternative or additional cost (such as a buyback or kicker cost), or a particular mode, was chosen for it; otherwise, the spell is cast as though it did not require those targets. If the spell has a variable number of targets, the player announces how many targets he or she will choose before he or she announces those targets. The same target can’t be chosen multiple times for any one instance of the word “target” on the spell. However, if the spell uses the word “target” in multiple places, the same object, player, or zone can be chosen once for each instance of the word “target” (as long as it fits the targeting criteria). If any effects say that an object or player must be chosen as a target, the player chooses targets so that he or she obeys the maximum possible number of such effects without violating any rules or effects that say that an object or player can’t be chosen as a target. The chosen players, objects, and/or zones each become a target of that spell. (Any abilities that trigger when those players, objects, and/or zones become the target of a spell trigger at this point; they’ll wait to be put on the stack until the spell has finished being cast.)
602.2b The remainder of the process for activating an ability is identical to the process for casting a spell listed in rules 601.2b – h. Those rules apply to activating an a bility just as they apply to casting a spell. An activated ability’s analog to a spell’s mana cost (as referenced in rule 601.2e) is its activation cost.