Dates and time in Chinese / Fechas y horas en chino

Months and days of the week do not have names in Chinese. They are numbered instead.
January is 一月 ( 1 月) and December is 十二月。
Monday is 星期一 and Saturday is 星期六.

The only exception is Sunday, which is called 星期天 or 星期日.  
Numbers are placed before 年, 月, 日 but after 星期.

From big to small

A rule in Chinese when talking about dates and time is to always go from the most general (big) to the most specific (small).  

DATES 

Start with the most general information (the year) and work your way down.

Structure
年 year > 月 month > 日 day > 星期 day of the week

Examples
二 零 一 四 年 一月 三 日 
January 3rd, 2014

Dates are also commonly written like this:
2015 年 1 月 3 日.
January 3rd, 2015

2016 年 1 月 3 日   星期日 
Sunday, January 3, 2016

TIME

Structure

Just like with dates, to organize how you talk about time, start with the larger element, here the period of the day. Going from largest to smallest is something you'll often find yourself doing in Mandarin. So, to talk about a given time, start with:
Period of the day > 点 (short for 点钟) hour > 分 (short for 分钟minutes > seconds

Examples
下午 五 点 三十 四 分 is 5:34 p.m.
现在 夜里 一点 一 刻。It's 1:15 a.m.

Useful vocabulary to talk about the time
+ 早上: early morning
+ 上午: morning
+ 中午: noon
+ 下午: afternoon
+ 晚上: evening
+ 夜里: night
+ 点 (short for 点钟): hour
+ 分 (short for 分钟): minutes
+ 刻: quarter of an hour
+ 半: half an hour
+ 差: here minus; to, as in 差 十 分 两 点: Ten to 2.