From my testing of combat discoveries:
- Choose an animal that gives the appropriate discovery.
- Make sure you're wielding the type of weapon needed by the move you're trying to discover.
- Make sure you're satisfying the IP requirements of the move you're trying to discover. You may need more IP, but it's definitely connected.
When you begin combat, the game decides if you will see a discovery, and the longer the combat goes on the more likely it is to "reveal" it. This happens per combat not per animal, so if you've fought an animal for 10 or so moves without seeing a discovery you should disengage and try again (if possible).
Because discoveries are generated per combat, if you can trap a fleeing animal you can engage it over and over to get multiple discoveries. We used this to get our whole village cleave in the first couple of days.
For chop - equip stone axe, make sure your deck includes take aim. Aggro an ant, take aim once or twice, then rotate through all your defences a couple of times until a combat discovery pops up. If nothing appears, kill the ant and find another one. Beyond this, it's luck. You can't force a new move to appear, but if you satisfy the requirements you can make it one of the possible discoveries.