Step 1: What day is today?

A: "Kyou wa nan nichi desuka?"
(What day today is?)
B: "Ju-ichi nichi desu."
(It's 11th.)

Sentences in detail:

"kyou"(today) 
"wa"(particle "wa")-If you would like to review this, go back to lesson 4. 
"nan"(what) 
"nichi"(day) 
"desuka" -"desu" is the verb which means "is" in English. "ka" is used for question sentences. In other words, if you put "ka" after "desu", you are asking a question. 
"ju-ichi"(eleven)-If you would like to review numbers, go back to lesson 3. 

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Step 2: Days of the Month
As you noticed, in Japanese, you can say "nichi" for day. Unfortunately, things aren't that easy, but there is a rule for you to remember easily. Here are the days of the month in Japanese.

Days of 1st~10th
"tsu-i-tachi"(1st) 
"fu-tsu-ka"(2nd) 
"mi-kka"(3rd) 
"yo-kka"(4th) 
"i-tsu-ka"(5th) 
"mu-i-ka"(6th) 
"na-no-ka"(7th) 
"yo-u-ka"(8th) 
"koko-no-ka"(9th) 
"to-u-ka"(10th) 
For these numbers above, I can't give a rule, memorize them!

Days of 11st~20th
"ju-ichi-nichi"(11th) 
"ju-ni-nichi"(12th) 
"ju-san-nichi"(13th) 
"ju-yo-kka"(14th) 
"ju-go-nichi"(15th) 
"ju-roku-nichi"(16th) 
"ju-nana-nichi"(17th) 
"ju-hachi-nichi"(18th) 
"ju-ku-nichi"(19th) 
"ha-tsu-ka"(20th) 
This is much easier than 1st through 10th because except 14th and 20th, you can use "nichi" for the day.

Days of 21st~31st
"niju-ichi-nichi"(21st) 
"niju-ni-nichi"(22nd) 
"niju-san-nichi"(23rd) 
"niju-yokka"(24th) 
"niju-go-nichi"(25th) 
"niju-roku-nichi"(26th) 
"niju-nana-nichi"(27th) 
"niju-hachi-nichi"(28th) 
"niju-ku-nichi"(29th) 
"san-ju-nichi"(30th) 
"san-ju-ichi-nichi"(31st) 
Again, this is very easy. I hope you find the rule for 21st through 31st. You're right. Except 24th, you can use "nichi" for the day. Is that easy? 

A: "Kyou wa nan nichi desuka?"
(What day today is?)
B: "(put the day of the month here) desu."
(It's ***.)

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Step 2: Days of the Week
In this step, you will learn the names of the days of the week. Let's start from Sunday!

"nichi-yo-u-bi"(Sunday) 
"getsu-yo-u-bi"(Monday) 
"ka-yo-u-bi"(Tuesday) 
"sui-yo-u-bi"(Wednesday) 
"moku-yo-u-bi"(Thursday) 
"kin-yo-u-bi"(Friday) 
"do-yo-u-bi"(Saturday) 
"yo-u-bi" means the day of the week.

A: "Kyou wa nan yo-u-bi desuka?"
(What day [of the week] is today?)
B: "*** yo-u-bi desu."
(It's ***day.)

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Half Time
Before going to the next step, I would like you to know the names of planets. Why? I don't know, cause planets are now officially kewl!

"sui-sei"(Mercury) 
"kin-sei"(Venus) 
"chi-kyu"(Earth) 
"ka-sei"(Mars) 
"moku-sei"(Jupiter) 
"do-sei"(Saturn) 
Did you notice that there are connections between the days of week and the names of the planets??  Well there is!

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Step 4: The Name of the Months
"Gatsu" means month

"ichi-gatsu"(January) 
"ni-gatsu"(February) 
"san-gatsu"(March) 
"shi-gatsu"(April) -Please do NOT say "yon-gatsu" instead. 
"go-gatsu"(May) 
"roku-gatsu"(June) 
"shi-chi-gatsu"(July
"hachi-gatsu"(August) 
"ku-gatsu"(September) -Please do NOT say "kyu-gatsu" instead. 
"ju-gatsu"(October) 
"ju-ichi-gatsu"(November) 
"ju-ni-gatsu"(December) 
Did you understand the rule? Since December is 12th month of the year, you can say "ju-ni-gatsu"("ju-ni" + "gatsu").

A: "Nan gatsu desuka?"
(What month is this month?)
B: "*** gatsu desu."
(It's ***[put month here].)

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Step 4: What date is today?
Now you know the days of month, the days of week, and the names of the month. This means you can say the date in Japanese. Please try this example.

A: "Kyou wa nan gatsu nan nichi desuka?"
(What date is today?)
B: "Kyou wa *** gatsu [put day here] desu."
(It's [put month and day here.)

A: "Anata no tanjoubi wa itsu desuka?"
(When is your birthday?)
B: "[put month and day here] desu."
(It's ***.)

"anata no" (your) 
"tanjoubi" (birthday) 
"itsu" (when)