What are the major characteristics of the settings of the play A Midsummer Night's Dream?

Since this is a play, we are not given too many details about the setting. However, we can make some inferences based on the action of the play.

Quince’s house is in the village. Because it is in the village, the craftsmen want to go into the woods to practice their play. They do not want anyone watching them. They want the play to be a secret, and also might fear ridicule since they are not the best of actors.

BOTTOM

We will meet; and there we may rehearse most
obscenely and courageously. Take pains; be perfect: adieu.

QUINCE

At the duke's oak we meet. (Act 1, Scene 2) 

The woods are very interesting. They are the home of the fairies. We know that the woods are near Athens. Also, the woods are going through a sort of artificial winter due to the argument between Titania and Oberon.

Therefore the winds, piping to us in vain,
As in revenge, have suck'd up from the sea
Contagious fogs; which falling in the land
Have every pelting river made so proud
That they have overborne their continents:
The ox hath therefore stretch'd his yoke in vain,
The ploughman lost his sweat, and the green corn
Hath rotted ere his youth attain'd a beard … (Act 2, Scene 1) 

The woods are also where Titania has her lair. She is surrounded by little fairy handmaidens who are invisible to humans. There are many mentions of flowers and other plants you would find in the woods. Once Titania and Oberon make up, the woods are green again and back to normal.

The craftsmen choose a spot to practice because it is green, as there is a patch of grass they can use as their stage.

Pat, pat; and here's a marvellous convenient place
for our rehearsal. This green plot shall be our
stage, this hawthorn-brake our tiring-house; and we
will do it in action as we will do it before the duke. (Act 3, Scene 1)

Unfortunately, this place is too close to Titania, and Puck has some fun with them. He changes Bottom’s head into a donkey's head and frightens the rest of the craftsmen. Then Bottom wanders toward Titania and Puck anoints her with the secret potion so that she will fall in love with him.

We do not have much detail about Theseus’s palace, except that it is described as a palace. This is where everyone meets at the end for the wedding and the production of the play-within-a-play. All of the lovers end up properly matched, and the fairies bless the palace.