Again: You DON'T need to spece words/ punctuation marks in Chinese writing!

今天的天气
今天天气很冷。你们那边今天的天气怎么样？  
（Chinese speakers please let me know if the last sentence is correct! 谢谢）

 The explaination maybe a bit busy for you,but please be patient :)

的 is a word to structure possessive relationships. It's used after noun.

e.g: 今天的天气 Today‘S weather

       我的名字  My name

        大学的学生 University'S students/ Students of the University

        玛丽的哥哥    Mary'S brother

about 的, it can be omitted in these 2 special situaion

1.  Relationship between people, and the possesive unit (Nouns before 的 must be pronoun words like My, her, his, their )

e.g. 我的妈妈  or 我妈妈      My mother

      他的老师   or 我老师     His teacher

2. Relationship between people and the group/organizations he/she/they belong, and the possesive unit (Nouns before 的 must be pronoun words like My, her, his, their )

e.g. 我的大学 or 我大学  My Uni

      他们的国家 or  他们国家  their country

      她的家     or  她家  her home

Actually here 今天的天气 and 今天天气 both are right. Now you must wonder why 今天天气 is also right because it's not in neither of those 2 situaitons talked above. Well, I will explain it :)

Here 今天的天气 means Today'S weather. But 今天天气 mean Weather today. Let me make examples to help you understand it

Today'S weather is cold.  今天的天气很冷。

The weather today is cold./Today, the weather is cold.     今天天气很冷。/今天，天气很冷。 

(In the second sentence, you can see that the relationship of TODAY and WEATHER is NOT a prossessive relationship.)

你们那边今天的天气怎么样？  How's today's weather THERE.  

Yeah, the translation in English here is a bit weird, right?  I translate it in strice way because I want you to notice the differences of 今天的天气 and 今天天气。 So actually you can just translate it as： How is the weather today THERE. 

I hope that you can get what I mean :)

今天天气

今天我们这儿天气很冷。你们那边天气怎么样？我的朋友们！ 
（Chinese speakers please let me know if the last sentence is correct! 谢谢）

今天 天气

今天我这里天气很冷。你们那边今天天气怎么样？
（Chinese speakers please let me know if the last sentence is correct! 谢谢）