Assuming 10 strength/14 dex/16 int, we get:
Longbow is +4 attack for 1d8+2(6.5) damage at 150/600 feet.
Rapier is +4 attack for 1d8+2(6.5) damage in melee (using tasha's rules to swap proficiency)
Short Sword is +4 attack for 1d6+2(5.5) damage in melee
Long Sword is +2 attack for 1d8+0(4.5) damage in melee
Firebolt is +5 attack for 1d10(5.5) damage at 120 feet
Shocking grasp is +5 attack for 1d8(4.5) damage in melee, and shuts down reactions (allowing you to flee and not get an opportunity attack).
- Missing sucks. When in doubt, bias towards accuracy.
- Against a low AC foe (13), the rapier/longbow does 7% more damage per round than the firebolt.
- Against a high AC foe (18), the longbow/rapier does 1% more damage per round than the firebolt.
- The short sword competes with shocking grasp similarly.
- The long sword option sucks.
That gap is pretty small between rapier/longbow and firebolt. So it ends up being a matter of style - does the player want to play a elf wizard who uses longbows, or shoots fire from her fingertips?
Unless the wizard becomes a bladesinger, by tier 2 cantrips pull ahead; you'll both probably boost int rather than dex, and they get a 2nd damage die.
If the player likes using weapon, the bladesinger is a solid continuation. At level 6, a 18 int/14 dex bladesinger on an AC 16 foe puts out:
- +7 to hit @ 2d10 damage firebolt (7.2 DPR)
- +5 to hit @ 1d8+2 damage arrow (3.5 DPR)
and can do both as an action. A non-bladesinger only does the firebolt.