I just finished taking a class on this, so this is pretty fresh in my brain. Please note that these are general findings about kids at this age, and of course all kids are different.
At age two, kids will put a few words together to create meaning, but not grammatically correct. For example, "Me want juice." Some common errors you will see at this age would be errors with overregularization: applying common semantic rules to everything, even exceptions. (Ex. "I goed to the playground.") Or overextension. (Ex. If they know a cat named Whiskers and see any other cat, "There's Whiskers!"). Similarly, there's underextension. ("These aren't my shoes; these are my boots.")
Kids at this age are not able to think abstractly, though it's been shown that infants under one year old can understand when someone is being mean vs. when someone is being kind. They tend to focus on the most visually striking feature of an object or substance and tend to disregard all other features. (This is called centering.) They don't quite understand identity constancy, which means they don't understand that a person's core self remains despite changes to external appearance. For example, if your dad puts on a dinosaur suit, he's a terrifying monster now, not your dad. They don't understand that other people have different perspectives than them (called egocentrism.) They believe that inanimate objects are alive (animism) and that human beings make everything in nature (artificialism). Kids at this age learn mostly by following modeled behavior, and they will often look to a trusted adult to know how to react to something. So if you yourself are afraid of spiders but you don't want your kid to be, model the behavior that it isn't scary, because the most effective way your child will learn what to think about a new object or activity is how you (trusted adult) react to it.
At this age, kids are learning to tackle autonomy. It might seem tedious and annoying to wait for your two-year-old to try to put their shoes on when you could do it in five seconds, but they are trying to learn to do things for themselves at that stage. Around age two, kids will start to develop self-conscious emotions like pride, shame, and guilt. These will develop for the next few years. At this stage they will begin to be able to obey your rules when you are out of the room. (Maybe not with 100% accuracy, but it begins taking form.)
Aggression is at its peak at two and a half years old, and they have not yet mastered emotion regulation at that point. Emotion regulation doesn't start to develop until middle childhood, starting around age six or so. At age two to four, kids tend to prefer collaborative pretend play, which is where kids work together to develop and act out scenes. Kids will start preferring to play with kids of the same gender as them around preschool. In that case, girls tend to play collaboratively, one on one, and boys tend to play in groups, but separately.
I hope this helps!
Sources:
The giant notebook full of notes I took on this psychology class
This article about 2-3 year old speech
This article about 3-4 year old speech
