How do I know what nut to get for training wheels?

In brief, you need to identify the major diameter and the thread pitch. That should be sufficient. Secondary considerations like the thread's form, its minor diameter, are less important. This also assumes the existing thread is not damaged.

1. You will need a vernier caliper - a ruler is not accurate enough, though a cheap plastic vernier may work.
   

2. You need a thread gauge. Bicycle threads are frequently Imperial (because reasons) but not always. And its possible that the wrong thread might be close enough to go on but not tighten properly. IE the difference between 24 and 26 TPI only shows up after a couple of turns of the nut.

The known sizes are:

Useful link: https://www.parktool.com/blog/repair-help/basic-thread-concepts

The other answer is functional, but buying tools to check threads is going way beyond what someone needs who just wants to put training wheels on a bike. Assuming you are in a North American country, head out to the nearest hardware store (Ace, True Value, etc.) or home improvement store (Lowes, Home Depot, etc.) and bring the training wheels with you. Its possible that the threads are destroyed on either the nut or bolt. The employees at these stores are always helpful and can find exactly what you need, or sometimes see issues that you can't. If employees aren't there or not helpful, there are usually nuts and bolts in the hardware section you can use to test out different sizes for fit.