As I understand the Fifth Edition rules, my player character may only have one spell that requires concentration active and in place at a time.

Levitate, invisibility, and magic weapon all require concentration.

So, preparing before that big final battle, my wizard may only have one of those spells active at a time, regardless of whether he is the target for all or for none.

You can’t concentrate on two spells at once.

Am I misreading the rules? Or is there a way my wizard can cast invisibility on the party rogue, magic weapon on the spear of the party fighter, and levitate on himself, and have all three active before they attack the Big Bad Opponent?

Use glyph of warding

If you cast glyph of warding and choose the option Spell Glyph:

You can store a prepared spell of 3rd level or lower in the glyph by
casting it as part of creating the glyph. The spell must target a
single creature or an area. [...] If the spell requires concentration,
it lasts until the end of its full duration. (PHB 245-6)

This will allow you to cast levitate and invisibility on a creature of your choice by casting the glyph on a book and setting the condition "target: the one who opens the book" or something similar. You will have to use your actual concentration for magic weapon, in your example.

The drawbacks are that this requires the expenditure of a level 3 spell slot, 1 hour of casting and 200 gp of materials per spell. If you know where the fight will happen (as the glyph cannot be moved (generally)) and prepare ahead of time, you might be able to spend the slots and time at an earlier date.

This may sound a bit cheesy, but the cost is really high if done on the spot and a risk to pull off or it shows proper preparation if done earlier (and it is not guaranteed than nobody will notice the inscriptions, depending on the site).

Note:

The description of the spell in earlier printings starts with:

When you cast this spell, you inscribe a glyph that harms other
creatures

But this has been changed in errata (ver. 1.22):

Glyph of Warding (p. 245). The first sentence clarifies that the
magical effect needn’t be harmful.

You can't.

As you've correctly quoted, you cannot concentrate on two spells at once.

As Levitate, Invisibility and Magic Weapon require concentration - a single wizard simply cannot do this alone.

However you could have all of these effects on you at once - by having multiple spellcasters help.

• Levitate can target any creature or loose object that you can see in range.


• Invisibility can be cast on any creature you touch


• Magic Weapon can be cast on any non magical weapon you can touch.

Therefore there is nothing stopping the collusion of multiple casters for these spells, as none of them specifically target "self". In fact with 3 spellcaster friends, they could cast these on you (and your equipment) then go hide, leaving you to not even have to worry about concentration checks when you get hit.