Learn English – Shorter phrase or word to say `the day before yesterday`

You could say "the 15th of June 2013 or any other saturday after that" if you mean that you could attend on the 15th or, any saturday that is after that.

If you write the date the English and not American way, you would need the words the and of in it. Including the year is not necessary if you are talking about a date in the same year. If the 15th of June will be later in the same year, it would be sufficient to say "the fifteenth of June".

As Nick Stauner's explained in his answer, both "take the word" was used to refer to "to accept their word for a conclusion of the issue." However, it might be beneficial to consider each of the involved sentences, one at a time. Here is the first one.

People want to use it to cook with and they're looking to take the word.

This is obviously related to the question "But how do you convince people that the water supply is, in fact, safe?". It means people want to use the water, and they want to be assured that the "water is safe".

We can understand this take the word by looking up the words take and word in a dictionary. For example, here are the related entries from Macmillan Dictionary,

take _{5 [transitive]} to accept something that someone offers you
word _{4 [singular/uncountable]} news or information about someone or something

_{Thus, we can understand "they're looking to take the word" as the people are looking to accept the news or information (in this case, the approval from the Health Department, according to a preceding paragraph in the transcript). Also note that this is not a common usage, as J.R. and others commented. Take someone's word for it is more idiomatic and recommended for learners.}

Here is the second sentence,

And I believe if they say to go ahead, I would take their word for it and I think the public, at large, will, too.

The phrase take someone's word for it is common enough that dictionaries would list it. Here is what I found in the same dictionary,

take _5a. to accept an explanation or something that someone says without discussing it or arguing about it
  take someone's word for it (=believe what someone tells you): You don't have to take my word for it – ask anyone.

_{This means that the mayor trust the Health Department, and if they say the water is safe, he would believe them.}