How do I catch 'critters'?

If you're fast, you can run up to frogs near some ponds and catch them on the go. That's also possible with lizards, but a lot more difficult.

But you're right, you can't simply waltz up to most critters as you'll scare them off. You need to sneak up to them slowly. Crouching helps, as does moving slowly. Remember that you have a sound meter that shows you how much noise you're making. Make sure to keep an eye on it and keep the line as flat as possible and you should be able to sneak up to bugs and catch them. You can easily catch 3-4 fireflies at night simply be crouching around them.

The noise meter is the purple line in the bottom right corner. The more erratic it is, the more noise you're making. Handy for sneaking up on enemies too.

My way of getting critters is that I've fully upgraded the Sheikah armor set, which gives a lot of stealth and movement speed at night. I can run straight up to a critter and it won't move for a second or so, giving me plenty of time to grab it.