For some of my build designs, optimization can come from versatility as opposed to consistent DPS. This build optimizes Cantrips and some spell-class diversity. At Wizard-1/Cleric-1, this build can don Chain Mail armor with a shield and Longsword/Longbow, have a pet Familiar, but also cast Bless, Charm Person, Sleep, or Tasha's Hideous Laughter as actions, Feather Fall or Shield as reactions, and even Healing Word or Sanctuary as bonus actions. The build also has 7 Cantrips with only 300 experience points invested. This is all before adding Bladesinger.
Standard-Array (Option 1): High Elf; Str 13, Dex 12 (10+2), Con 12, Int 15 (14+1), Wis 15, Cha 8
Standard-Array (Option 2): High Elf; Str 13, Dex 12 (10+2), Con 12, Int 16 (15+1), Wis 14, Cha 8
Point-Buy (Option 1): High Elf; Str 13, Dex 10 (8+2), Con 12, Int 16 (15+1), Wis 15, Cha 8
Point-Buy (Option 2): High Elf; Str 13, Dex 11 (9+2), Con 13, Int 16 (15+1), Wis 14, Cha 8
In this build, the Wizard starts at Level 1 (L1), but switches to Life Cleric at L2 and then back to Wizard for Bladeslinger at L3 and thereafter. An alternative to this build would be to start as a Life Cleric at L1 and then proceed to choose Wizard at L2 and after; for the Simple weapon proficiencies and with the option to choose Wisdom-backed skills over Intelligence-backed ones.
The build can don Studded Leather with no shield for initial encounters. It only takes one action to don a shield when you have run out of Bladesongs, but it's almost-always worthwhile to don Chain mail-and shield when you run out of Bladesongs rather than complete a short rest, especially if you DM only allows short rests twice a day (or less) or places other restrictions on them.
Optionally, the Bladesinger could start adventuring with Chain mail-and Shield in order to save the two Bladesongs for miniboss and boss encounters, doffing the Chain mail-and Shield for the Studded Leather at the appropriate time. In some cases, I think this works well at lower levels until Intelligence is raised at L5 and L9, especially before L11 when the Song of Defense arcane-tradition feature is gained. Before Song of Defense, there really isn't a strong desire to activate Bladesong during combat, opting instead for its out-of combat effects (i.e., the ones that aren't buffed by Intelligence, such as the movement-speed increase, the advantage on Acrobatics checks, and the Extra Attack). Note that the Extra Attack applies even when Bladesong isn't active.
Heavy Armor and Shield can be worthwhile at higher levels as well as low ones, especially if the character adds in magical items that artificially raise Strength and Constitution. Not to mention the bonuses that magical armor and shields give.
While I only chose 1 level of Life Cleric, this build can still raise both Int and Wis to 20 even if 3-5 levels of Cleric are taken. With 3 levels of Cleric, Spiritual Weapon becomes available and combines nicely with a high-level Spell Slot. In my version of the build, I'd probably opt to raise Constitution to at least 16 and only raise Wisdom as high as 16 (or even leave it at 14 if one of the Option 2s are chosen above). Usually there will be no need to raise Strength above 14, except maybe to 15 if Plate is desired (and no strength-raising magical items are ever attuned) -- although if the character relies on consistent, strong melee attacks then I would likely prefer to raise Dexterity, but I think this makes the build too-MAD.
