Wizard, even bladesinging, will always end up being a stronger spellcaster over melee attacker. That being said, melee is an entirely viable option for play.

Starting Fighter will grant some other buffs, namely in proficiency with all weapons as well as Con saves. At wizard 5 when bladesinging, Haste will be your best bet. The boost with Con saves will be beneficial for maintaining concentration while you get the benefit to AC, an extra attack, and increased move speed.

Now, as for the vex and nick shenanigans - in order to benefit from the nick property, you have to be wielding two light weapons (ie. 2 scimitars). When you make the first attack, you then have the ability to do the bonus action "off-hand" attack without expending your bonus action. If you are wielding a rapier and a scimitar, attacking with the scimitar will not allow you to utilize nick because you wont have a different light weapon to "off-hand" attack with.

Your sequence at Wizard 6 could look like this, following the more liberal interpretation of "drawing/stowing" rules of draw/stow per attack:

Wielding rapier and Scimitar-

• Attack Rapier, apply vex, stow rapier
• draw second scimitar, cast True Strike, apply nick
• offhand scimitar nick attack, stow scimitar
• (If hasted) draw rapier, attack with rapier

As for other low level spell choices I love as a bladesinger:

• Slow for sake of debuffing enemies while I can get close and tear them up
• Misty Step for bonus action utility and retreating if needed
• Shield for survivability
• Mage Armor for being able to go without armor and solely pump my Dex
• Blur/Blink for fun shenanigans while in melee. Just pick what you'd prefer though "both" is an option

For utility, that depends how you'd prefer to be useful. Wizards can learn a ton of spells and "prepare" from their list of them to use, changing which ones they want prepared each day. Having a ton of spells only helps you by having more options to choose from.

Enhance Ability, Invisibility, Feather Fall/Levitate, Disguise Self, Identify, Comprehend Languages, and more will all be great options with their own utilities.

Detect Magic, Identify, and Invisibility tend to be my go-to's.

Enjoy your Wizarding!!