The Starter Decks (Yugi, Kaiba and so forth) have simple cards, a random smattering of effects and so on to ease you into the game. The Structure Decks are themed, and often contain more complicated effects, as well as a number of combos the Starter Decks almost completely lack. The booster sets contain random sets of cards, with something of a tendency for newer sets to contain more complicated effects, akin to the Structure Deck (with the older sets tending to contain the simple and powerful stuff). The newer stuff in general relies on more extensive knowledge of the game, such as knowing the steps of a battle, and how to combo.
It all depends on how much you know of the game, really. If you just know basic gameplay (which sounds likely, judging from what you've been saying), then the Starter Decks and older booster sets might be more suitable, to give you more experience in playing the game. Then you can build up by looking at other cards lying around, seeing and learning how they work, and developing your knowledge of the game. You could even use Ronin or Deck Studio for such research, which saves you having to buy anything.
I'd suggest, even, that you forego getting the Yugi and Kaiba Starter Decks, and instead purchase their Evolution counterparts. They're still simple enough (and are at least more playable than the Joey or Pegasus Decks - but you can always get these as well if you like). Then maybe some packs of a few booster sets, to introduce you to some cards you might not have expected, and when you're comfortable with that, maybe a Structure Deck. Get one you're interested in, not necessarily the latest one - you can always ask, and the latest rulebook is online anyway (see below).
Link 1: http://entertainment.upperdeck.com/yugioh/en/gameplay/rulebook/06_SD_Beginners_Guide_EN.pdf
This is the Beginner's Guide. Have a look at this first. It contains all the basics. If you're happy with it, then you know basic gameplay, and should be able to use the Starter Decks. But before you so, also have a look at...
Link 2: http://entertainment.upperdeck.com/yugioh/en/gameplay/demo/default.aspx
Here's an interactive demo that covers a bunch of basic information, such as how to play cards. I remember it being rather informative, and it helps to show how the cards move about, because the written material (links 1 and 3) might not be too clear on this.
Link 3: http://entertainment.upperdeck.com/yugioh/en/gameplay/rulebook/rulebook_v06_EN.pdf
This is the official rulebook, the one EmeraldDragon (above) mentions. You'll find a rulebook in every Starter Deck or Structure Deck you buy, and there are a number of versions of it. Use the latest version (version 6, I believe), because it's the most correct and informative. One you've read it all and understand at least most of it, you're in good stead, and you can probably use any cards with relative ease. Some you'll have difficulty with, I'm sure, because everyone does (Union monsters, Spirit monsters, Spell Counters and Monster Tokens are all concepts described perfectly well by their associated cards, but are not properly introduced in the rulebook), but if you have any difficulty don't hesitate to ask (preferably here, because you can find a lot of misleading information in other places).
