Learn French: Tu devrais laisser tes chaussures dehors. - You should leave your shoes outside.

Devrais is the conditional form of the verb devoir (must / to have to). While Tu dois gives a strong command like 'You must', using the conditional Tu devrais softens it to 'You should'. It is the standard way to give friendly advice in French.

Why isn't the second verb conjugated?

In French, when two verbs are used together to express a single idea, only the first one is conjugated. Since devrais is already matched to Tu, the second verb laisser (to leave) stays in its infinitive (dictionary) form.

Could I use partir instead of laisser to say 'leave' here?

No, because they mean different types of 'leaving'. You use partir when a person or vehicle is departing a place. You use laisser when you are leaving an object (like tes chaussures) behind.

Do I need to put a word like à or en before dehors?

No, dehors is an adverb meaning 'outside' or 'outdoors', so it stands alone perfectly here. You don't need any extra prepositions.