Can you choose a new target for a spell after your opponent counters the original target?

When a spell is cast, targets are chosen at that time and locked in (CR 601.2c). You cannot simply choose new targets whenever you feel like it — targets can only be changed if a spell or ability specifically allows it, such as spells that say 'change the target of target spell' (CR 115.7).

If your opponent 'counters a target' — meaning they use an effect that removes or destroys the targeted permanent before your spell resolves — your spell doesn't automatically get a new target. Instead, when your spell tries to resolve, any targets are checked for legality (CR 608.2b). If all targets are illegal at that point, the spell is countered by the rules and fails to resolve.

There is one relevant exception: if only some targets are illegal, the spell still resolves but only affects the remaining legal targets. This is different from choosing new ones — it's simply ignoring the illegal ones (CR 608.2b).

Importantly, 'countering a target' is not the same as countering the spell itself. Effects like Counterspell counter the spell directly. Destroying the target (e.g., killing a creature your opponent is targeting with Doom Blade) does not counter the spell — it just makes the target illegal upon resolution.

Example: You cast Lightning Bolt targeting your opponent's Grizzly Bears. They cast Giant Growth on the Bears in response. You cannot pick a new target for Bolt; it will still resolve targeting the Bears (which may now survive due to toughness), unless the Bears leave the battlefield entirely, in which case Bolt is countered by rules for having no legal targets.