If I have a counterspell in my hand, and my opponent chooses to cast an instant or sorcery, can I wait to see what his spell 'targets' before I choose to counter it or not?
Yes, you will know all targets of a spell as soon as it is cast.
From the Comprehensive Rules:
601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. Casting a spell follows the steps listed below, in order. If, at any point during the casting of a spell, a player is unable to comply with any of the steps listed below, the casting of the spell is illegal; the game returns to the moment before that spell started to be cast (see rule 717, “Handling Illegal Actions”). Announcements and payments can’t be altered after they’ve been made.
601.2c The player announces his or her choice of an appropriate player, object, or zone for each target the spell requires. [...]
601.2h Once the steps described in 601.2a–g are completed, the spell becomes cast. Any abilities that trigger when a spell is cast or put onto the stack trigger at this time. If the spell’s controller had priority before casting it, he or she gets priority.
Note that depending on the card-specific wording, there are some spells and abilities that have choices that are made when they resolve, not when they are cast. Once a spell or ability starts resolving, there is no way of countering it, so in those rare cases, you have no way of knowing for sure what it will do.
For example, Hex Parasite's ability lets you choose the number and kind of counters to remove when it resolves, which matters when the target has more than 1 kind of counter on it.
Another side effect of this rule is that there is usually no way of preventing abilities that trigger on its permanent being targeted from triggering, aside from hexproof or shroud. Once you target such a permanent (usually a creature), the ability will trigger, even if you counter the spell that triggered the ability.
Yes, you may wait to see what the spell targets.
Instants and Sorceries targets are declared when the spell is cast, and are checked again when the spell resolves to ensure they are still legal targets.
114.1. Some spells and abilities require their controller to choose one or more targets for them. The targets are object(s), player(s), and/or zone(s) the spell or ability will affect. These targets are declared as part of the process of putting the spell or ability on the stack. The targets can't be changed except by another spell or ability that explicitly says it can do so.
114.1a An instant or sorcery spell is targeted if its spell ability identifies something it will affect by using the phrase "target [something]," where the "something" is a phrase that describes an object, player, or zone. The target(s) are chosen as the spell is cast; see rule 601.2c. (If an activated or triggered ability of an instant or sorcery uses the word target, that ability is targeted, but the spell is not.)
608.2b If the spell or ability specifies targets, it checks whether the targets are still legal. [...]
A card that doesn't use the word target in its rules text, or in its keyword ability (Aura, Equip, etc.) does not target. With a card like Kindle the Carnage/Living Wish, the amount of damage it deals (or the card it puts into hand) is decided during the resolution of the spell, so you could not wait to see how much damage the spell would deal (or which card is chosen) before deciding to counter it.
Similarly, with a permanent card like Oblivion Ring/Clone, that has a triggered ability or choice that is made when the object enters the battlefield, you cannot wait until the choices or triggers of those cards occur. If you do, that would mean that the spell had finished resolving and those spells are now objects on the battlefield.