Does copying a spell with Fork also copy the targets of the original spell?

When you cast Fork targeting an instant or sorcery spell, Fork creates a copy of that spell on the stack. By default, the copy has the same targets as the original spell (CR 707.10). However, you are not locked into those targets — Fork's own text explicitly allows you to choose new targets for the copy, and CR 707.10 confirms that when a spell is copied and the copy grants the option to choose new targets, the controller may do so.

This means you get a choice at copy-creation time: keep the same targets or redirect the copy to entirely new legal targets. You cannot mix and match individual targets; the rule lets you change any or all targets, but only at that moment (CR 707.10, 114.6).

The original spell is completely unaffected by Fork — it remains on the stack with its original targets and resolves normally unless countered or otherwise disrupted. Only the copy is subject to your new target choices.

Example: Your opponent casts Lightning Bolt targeting your 3/3 creature. You cast Fork targeting that Lightning Bolt. A copy of Lightning Bolt is created; you may keep your creature as the target (odd choice) or redirect the copy to hit your opponent directly for 3 damage. The original Lightning Bolt still resolves and deals 3 damage to your creature unless you deal with it separately.

Note that the copy must still obey all targeting rules — any new target you choose must be a legal target for Lightning Bolt (CR 114.4). If no legal alternative target exists, you are stuck with the original target or the copy is simply a copy with the same target.