Generally, you need to find a fish that's big enough to break it. If you're trying to break a Halcyon, I think the lowest fish that can break it is a Gugru Tuna. You'd need a minnow or slices of sardine for that, probably. (Note: if you're trying to break it several times, and I suspect you are, you'll lose the minnow if the rod breaks, so you might want bait instead of a lure.)
When you see a message that says "Something caught the hook" look for 3 exclamation points after the message. That means it's either a big fish or a monster. If it says "You have a terrible feeling about this one," then it is definitely a big fish. To break the rod, you must tire the fish out completely and try to reel it in. You might just snap your line, or you might break the rod completely. The bigger the fish, the more likely it is to snap the rod. However, bigger fish are harder to reel in, so if your skill level is low, you might have trouble wearing bigger fish out in time.