Learn Russian: Не надо покупать лишние вещи для дома. - No need to buy extra things for the house.

After не надо (no need to / shouldn't), Russian strictly requires an imperfective infinitive, even if you are talking about a single occasion. The combination of a negative word and a word of obligation emphasizes the idea of avoiding the action entirely.

What exactly does лишние mean here? Does it just mean 'extra'?

Лишний translates to 'extra' but usually carries a negative tone, meaning 'superfluous' or 'unnecessary' (literally 'things we don't need'). If you want to say 'extra' in a positive or neutral way, like an extra ticket, you would use дополнительный.

Why is вещи used here, and what case is it in?

It is in the accusative plural because it is the direct object of покупать (to buy). Because вещь (thing) is an inanimate noun, its accusative plural form вещи looks exactly the same as its nominative plural.

Why do we say для дома instead of using the dative case?

The preposition для (for) followed by the genitive case дома shows the intended purpose or use of the items. You generally use the dative case without a preposition when an action is directed at a recipient (like giving a gift to a person), but for objects intended 'for' a place, для is correct.