Use the preposition 'into' to talk about movement from the outside to the inside of something:
He jumped into the ocean.
The baby climbed into the cabinet.
Use the preposition across to talk about:
1. things that go from one side to another:
The man is jumping across the gap.
2. things that are on the other side of something:
The building is across the street from us.
Inside is like 'in', but more specific to enclosed spaces (= a space with sides, walls, a ceiling):
There are many gifts inside the mailbox.
The dirty clothes are inside the washing machine.
We do not say that someone is inside a country/a forest/a park, because these places are usually not physically enclosed. We say in a country/a forest/a park.
Use the preposition outside to talk about people/things/animals that are not inside something:
Please leave your shoes outside.
There is a dog outside the shop.
You will often hear 'inside' and 'outside' used by themselves as adverbs. For example: "It's raining. Please come inside (the house, car, etc.). Don't stand outside."
Use the preposition beside or next to to talk about someone/something that is very close to someone/something else:
A woman is sitting next to a man. / A man is sitting next to a woman.
The cups are beside the sink.
