TWF With Punching?
 

I'm building a character right now, and I want him to punch people, and in order to make him punch a lot without multiclassing into Monk, I'm taking Two-Weapon Fighting.
The stats for two-weapon fighting don't show up with Unarmed Strikes.
Is there a way to show it on my weapons page, or do I have to write a note to myself?

What exactly don't you see? The effects of the 2Wpn feat are automatically applied to your character. This is the same way it works if you actually had weapons instead of unarmed strikes. (I've checked and it works properly)
The bonus/penalties that the feat applies are both mentioned on the feat tab, and a PDF of the character.

Just check both hands (hand 1, hand 2)

Note that a single level of Monk gives you the ability to use feet, knees, elbows, and even head butts, without penalties. This can be a big advantage when you want to hold something in your hands. You might still want the TWF feat tree, but it provides flexibility without much cost (d8 instead of d10, but +2 on all saves; just don't take Monk at 1st level, take it at 2nd or later).

Let's say I have a level one Cavalier with 16 Strength.
That's an unarmed attack of +4 1d3+3.
I take Two-Weapon Fighting at level one, and it stays at +4 1d3+3, with no way to fiddle it into +2/+2 1d3+3.

I'm wondering if I can just give myself an extra unarmed strike so I can just highlight them both, but there's no option to do so in the Adjustments tab.

At +2/+2 1d3+3 is wrong. By default if you check the main hand and 2-hand for unarmed strike you get: unarmed strike +2 (1d3+3 nonlethal)/1d3+1 nonlethal) which indicates two attacks at +2 doing different damage.

As Minous says - You should see Melee: +2 with the two damages listed. Without selecting BOTH 1st and 2nd (both hands) you will see +4 / 1d3+3 with Two Weapon Fighting selected. It sounds like you haven't selected both hands for Unarmed Strike.
Here's a screenshot of the weapon tab - I made a 1st Fighter since I don't have APG.

Huh, I didn't realize that I'd need Double Slice to offset the damage loss.
But I still have a problem with there being no offhand stats.
Here, for example, this is Officer Getskilled, an immortal mountain-biking vampire from the future.
He is a level-six Cavalier, as you can see.
He has Improved Unarmed Strike, Improved Grapple, Two-Weapon Fighting, Improved Two-Weapon Fighting, and Double Slice.
With ITWF, and selecting his brass knuckles, he has an attack of +7/+7/+2/+2 1d3+3.
But if I do it with his unarmed strike, it's still +7/+2 1d8+3.
I have not found a way to add that second attack bonus, or even a way to just brute-force an extra unarmed strike onto the file.
The way I'm seeing it right now, the only way I can make this work the right way is to just rename some weapons and use them in punching's place.

I'm going to guess that TWF means "two weapon fighting" and it doesn't apply to natural attacks. Hence, you get extra attacks with brass knuckles but not unarmed.

This is starting to sound like a bug in the PF data file, and should be reported to LW.

Using TWF with unarmed strikes was answered in an Paizo FAQ in 2013:
https://paizo.com/paizo/faq/v5748nruor1fm#v5748eaic9qie

Unarmed Strike: Can I use two-weapon fighting to make two unarmed strikes in one round?
Yes.

I image the actual number of attacks a character would receive would depend on it's BAB and other feats (ITWF, GTWF, etc.). But at the base line with a 1st level character who selected TWF (without Improved Unarmed Strike), they should have 2 unarmed strikes, though those strikes would be nonlethal and provoke attacks of opportunity. And those attacks would be at -2 each.