How to express a date in Chinese (lunar) calendar?
You add "农历" (means old/lunar calendar) add the beginning, and the first month is called "正(zhēng)月". The last month is called "腊(là)月". Leap month (a whole month is added) is prefixed with "闰(rùn)". Other months are read just as in the calender used now. The 1st to the 10th day is prefixed with "初". Other date is read out just as you read out a number. "号" "日" or any other characters is never added after the date. When "正月" "初" is used, "农历" can be omitted.


  E.g.: The Spring Festival is on (农历)正月初一
  
  1.10 : (农历)正月初十
  
  8.15 : 农历八月十五
  
  For leap month: 闰四月初六


The last day of a year can be called "大年三十" or "年三十" (or sometimes simply "三十". This can be used no matter the month has 30 days or not). The first day of the year can be called "大年初一".

I'm not quite sure what will be used when a leap month is added after the first month. I would prefer "闰一月" instead of "闰正月", but not sure about the common practice. Also, it is unknown to me how to read the leap month added after the 12th month. It is rare, and the we don't often used the lunar calender now.