Generally I am reluctant to answer anonymous enquiries. Since I've already started, here it goes.
Out of the three question, two have been answered:
> Q1. Do we always need an Internet connection?
> Q3. Which Moodle Version is best for us?
The latest: 3.5 today, 3.6 tomorrow.
The third question is still hanging:
> Q2. Is it something someone with little IT experience can install?
It depends - on how little the IT experience is and how unusual your case is. The most usual case, running a packages Moodle site in the Internet is a one-click affair if you go to a mass-hoster. Your case is unusual. And I'm sorry to say, you are also making incoherent statements.
Look at these:
> We have five labs and each lab has a server without internet. I'm not sure if they are internally connected.
This is a critical piece of information.
> But I if I could load Moodle on each server that could be a start.
A bad start. Apart from maintaining five sites instead of one, a Moodle site is not something you can jump from one to the other. The site contains user specific data, which is needed the next time the user pay a visit.
> I think we need internet for Moodle Cloud. We have desktop computers without internet in the labs. This is why we need a local server
So the Cloud is something you can reach without the Internet. (Notice the uppercase Internet.)
> We are trying to get our company to purchase tablets for the weaker students.
Are you in California. One iPad a student was supposed to upgrade the education system of the whole state!
> I suppose a internet connection would be needed then
So you are ready to give up on this:
> we don't want students going to others sites while online so having Moodle locally is best for us.
??? And more...
I suggest a) have a clear idea what you want to achieve b) decide finally on the infrastructure. Post those two things in this forum and the Moodle experts will tell what can be done.
The answer to your next question, How?, depends on the What you can do.