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:
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.
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.