Capitalize the days of the week and the months of the year because they were (mostly) named after people. We don't capitalize seasons because they don't come from names unless they are part of the name of an event.
Monday (Moon's day)
Tuesday (Tew's day - a Norse god of war)
Wednesday (Woden's day - a Norse god who guides souls)
Thursday (Thor's day - another Norse god, the chief god)
Friday (Frige's day - another Norse goddess - Are you seeing a pattern yet?)
Saturday (Saturn's day - a Roman god - What's he doing here?)
Sunday (Sun's day)
January (Janus - Roman god of gates and doorways)
February (month of februa - purification)
March (Mars - Roman god of war)
April (Aphrodite - Greek goddess of love)
May (Maia - Roman goddess of spring)
June (Juno - Roman goddess of marriage and women)
July (named after Julius Caesar, the emperor who began a revision of the Roman calendar)
August (named after Augustus Caesar, who finished the calendar revision)
September (seventh month)
October (eighth month)
November (ninth month)
December (tenth month - Obviously things have changed since they were named)
Seasons are only capitalized if they are part of the name of something.
Fall Festival, Winter Ball
Always capitalize the names of holidays.
Memorial Day, Flag Day, Easter, Fourth of July, Yom Kippur
Always capitalize the abbreviations b.c., a.d., c.e., and b.c.e. Usually these abbreviations are capitalized but at the height of lowercase letters. Some computers will do this automatically. You can do it manually by changing the font size. It is okay to leave it the capital height; it just may look strange to you.
Do not capitalize a.m. and p.m. In the past, publishers used a.m. and p.m. or am and pm. Now, most use a.m. and p.m.
We will leave tomorrow at 9:00 a.m.
The Roman Empire began in around 27 b.c.e. and lasted until a.d. 476.
(Note that a.d. [Anno Domini] precedes the date.)