For words indicating general direction, American English always uses the -ward suffix for both adjectives and adverbs: forward, eastward, toward; British English uses the -wards suffix only for adverbs.
Cardinal directions (north, east, south, west) and their compound forms take the lowercase unless used as proper nouns: He went north, It’s southeast from Quebec, The Pacific Northwest, East Germany, Northern Ireland. Do not capitalize the words when referring to a general area: eastern United States, south Vermont. While some locations may be easy to judge due to popular use (Southeast Asia) or a distinct border (West Germany), other locations may require familiarity with the region (Southern California) and should be capitalized following style guides or dictionaries.
For people, use the lowercase unless the words refer to specific social, political, or economic groups: southerner (of a country), Southerner (rebel of the American Civil War). The American North–South divide may lead to debate with terms like southern hospitality, southern gentry, southern rock, etc., but since none of these refer to specific titles, none should be capitalized. Some style guides, magazines, and regions may disagree; in such cases, use whatever capitalization brings you the least trouble.
For -ly endings, use the standard spellings northerly, easterly, southerly, westerly.