The Miniatures game was a separate product from 4e but it came with 5 minis and a large dragon so it used to be a good value. However, some of the original physical items are very expensive to buy nowadays. And keep in mind that some evolved as the game mechanics evolved to be much better than the originals. So the earlier in the production cycle something was produced the greater the chances that it has some clunkiness for roleplay aspects, but is perfectly OK for miniatures play.
The Initial Rules 4e Starter Set comes with a quick adventure, a quick primer of rules and some token cards for monsters, NPCs, etc.
Essentials Starter Set was the revamp of the rules for the Essentials product line but even this product suffered from being produced before the rules were completed, so it differs from the final Essentials product line rules. The quick adventure is OK, and the rules are solid but do not match products that came after. Character creation is done in a follow your path format with the path followed leading you to a character choice (Wizard, Fighter, Rogue or Cleric). And the product comes with power cards for those character classes.
Miniatures Starter Set is the miniatures combat game system. A simplified format where miniatures have point values, etc. The miniature cards resemble the monster cards from the Adventure System Board Games, but are not suited for use in that game. This game did not receive much support in this particular incarnation and disappeared quite quickly from the product landscape.
So the suggestion of using the free quick starter and Keep on the Shadowfell to get your feet wet is at least an OK one. Keep in mind that Keep on the Shadowfell is, IMO, one of the worse ways of introducing anyone to the 4e roleplaying game, but as a quick miniatures romp it is serviceable.
If you really want to take a look at the game for roleplaying purposes I'd rather recommend using the adventure Reavers of Harkenwold, which is part of the
Essentials DM Kit, as a good starting point.
If you are not interested in the roleplaying aspects of the game almost any version of 4e will provide a very solid framework for miniatures game play. If you are interested in the roleplaying aspects opinions differ as to what products provide the best support. I'm a proponent that roleplay aspects are something the group works out so for me any of the products was quite efficient, though skill challenges in the original DMG were presented pretty poorly.
