singsong
What species do you have? Different ones have somewhat different behavior.
Fluffing and shaking (called "rousing" by falconers) helps get the feathers in proper position. It is usually a sign of comfort, and is frequent after they bathe; they usually combine rousing with wing-flapping to dry off more quickly. If they rouse continuously without being wet for 5+ seconds while leaning forward or picking at themselves, there may be something bothering them (mine do this when another bird poops on them).
Beak-fencing is a mild form of aggression.
Preening birds oil their feathers to keep them clean. Often a bird will preen its body feathers with its beak, but use its feet to scratch its head and neck. They take oil from the preen gland on their rump and spread it onto their body feathers using their beak, or use their beak to get oil and then scratch their beak and then their head with their feet.
Allo-preening is when one bird preens another, and is a sign of bonding. They don't use oil for this.
Pecking is a sign of aggression, but is sometimes used by singing males to get the attention of another bird.
Plucking can be a result of stress/overcrowding, nutrient deficiency, pain (if done to oneself), too much blue lighting, or lack of nesting material.
Nesting calls are used when the bird finds a potential nest site or enters the nest. It seems to attract the mate to the nest site.
Contact calls are normal chirps that alert the birds to each other's presence. They can also be loud, drawn-out, or screamy when one bird is calling for another it thinks is lost.
Alarm calls can alert birds to the presence of a predator, and sometimes the type (flying, walking, etc).
While finch behavior is similar among species, calls can be quite different. Here's what my societies sound like:
Contact calls: one-part meep/mreep/cheep calls for males, two-part cricket-like calls for females. Much louder when stressed.
Alarm calls: quiet chip for low- to mid-level stress, somewhat loud 2-part chip for areal predators. Single chip results in looking around, raising head, and smoothing feathers, while double-chip results in flying for cover and freezing in place, for up to a few minutes
Nesting calls: whining, nasal mrrr, often repeated in clusters of 2-3
Birds of the same species, sex, and bloodline can still have different voices, so pay attention to the structure and context of the noise, rather than its specific tone when trying to figure out what it means. I know I've made that mistake before (three males had different stressed contact calls than their father and previous birds I had; I thought they meant different things, but that turned out wrong).