Ok, so you're looking at getting a handle on the basics; apparently coming from a 2nd edition campaign. We should be able to help you out there. Here's my first-pass at your questions, others will undoubtably clarify and correct as needed.:
>what is a partial action? What is a full action? How many of each
> can you have in a round? is it 1 full and that's it or is it 2
> partial?
Keep in mind that there are exceptions and special cases to pretty much every rule, but here are the basics:
A Standard Action is a "normal" action like swinging a sword or casting a spell under normal conditions. In a round, a character can generally move (up to their movement rate, which is 30ft for humans in no armor) and do one standard action in a round.
Now, a Partial Action is a Standard Action without the move. So you can stand still and cast a spell as a partial action (whereas you could move and cast a spell as a full round action).
You rarely "voluntarily" do a partial action; they only come into play in situations where you would do a standard action but cannot for some reason (like being surprised, or slowed, etc.). Haste allows an extra partial action in your round; and you can only "ready" partial actions. Those are the most common places for these to come up. When they do, just treat them like what you would do in a normal round, but without being allowed to move.
> How many free actions in one round?
As many as your DM will allow. Speaking is a free aciton but you really shouldn't speak more than 6 seconds worth of dialog. Usually, you can do 2 or 3 free actions without drawing the ire of your DM, but it is up to him in the end.
> How many actions does movement (moving your full base movement) constitute? Can you move and cast a spell? If so how far if the spell is a partial action (not a whole round spell)?
In a round, you can move your speed and performa a Standard Action. You can cast as spell as a standard action so yes, you can move and cast in a round (for most spells). Note that you can also move your speed as a standard action, which means you can move, and then move again in a round if you do nothing else.
Also, you can substitute a "Move Equivalent Action" for a move. For example, you could get something out of your belt pouch instead of moving. And as a special case, if you do move-equivalents instead of actually moving, then you get a free 5ft step.
> How are rounds divided? Is it 10 segments to a round or what?
There are no segements in 3rd Ed. A round is 6 seconds. A full round is 6 seconds. A partial round is 6 seconds. A spell that lasts 10 rounds lasts 60 seconds.