Those four expressions indicate conjecture, but many people do not have a clear idea of the differences betweem them.
「ようだ」 is an expression which is usually based on what the speaker see or saw. The expression involves the speaker's reasoning precess based on firsthand, reliable information and his knowledge. So it's subjective, and also the degree of certainty is the highest of the four expressions compared here.
「だろう」expresses the speakers' conjecture and is often used for uncertain matters like weather.
「らしい」 is generally used when the speaker make a conjecture based on some information which he has heard, read, or seen. What he has heard is especially important and this expression is used when the conjecture is based on more reliable information. Compared with「そうだ」, this expression is more objective and the speaker is less interested in the matter he is talking about.
「そうだ」 Although this expression is based on what the speaker sees or feels, it is merely his guess and degree of certainty in his statement is fairly low.