All you need to play 5e is the Basic set, some pencils and paper, and the standard 7-die RPG dice set.
The basic set player pdf has the four core classes, fighter, wizard, cleric & rogue featuring the most simple of their sub-class options allowing player to easily slide into the game and learn it. Additionally it covers the various pantheons of gods in the many D&D settings (as well as historical polythestic belief systems such as those held by the norse or the egyptians) and includes information on the 5 factions that NPCs and PCs can join and interact with.
In the DM pdf there is 63 pages, most of which is a fairly decent monster appendix for you to use to fill up your dungeons and towns and wild areas wit foes for the party to fight. It breaks down how to understand and use the monster stat blocks as well as what various monster abilities and monster types mean within the combat system as well as the default story setting for the game. Furthermore there is a section on how to build appropriate combat encounters for your party and a short section on the use of magic items within the game along with a small selection.
You can add options and complexity overtime by picking up the published books 1 at a time.
While the PHB, MM, and DMG make up the traditional trio of core books for 5e, you in no way need the 3 of them all at once to utilize any one of them. All 3 could be used individually with the materials in basic set to enhance your game. If you pickup basic and your players like it, I would suggest picking up the books in order from PHB -> MM -> DMG. The PHB itself will expand both your player's options as well as your materials as a DM (though on a much smaller scale for the DM). The MM will give you oodles of creatures to populate your world at various difficulty levels and provides a wealth of background information about the creatures helping you to roleplay them as well as providing inspiration for world building. Finally, the DMG offers a ton of optional rules, guidelines for customization, lots and lots of magic items, as well as a plethora of tables you can use for ad-hoc creation in your game on the fly.
Both Grid & Minis based combat and Theater of the mind/WYSIWYG styles of combat are supported by 5e
Spell descriptions, creature speeds, and ranges are all given in terms of feet. This allows for a more narrative approach to combat such as theater of the mind where all of it is run in heads of the players' and the DM, maybe with some scrap paper used to help illustrate rough positioning. It equally lends itself to using a ruler for WYSIWYG wargaming rules as well as simply dividing the distances by 5 to make use of a hex or square grid map. One of 5e's stated design goals was that it have lots of optional rules and approaches to allow you to play the game the way your table would prefer it to be played (and also to broaden market appeal).
Additionally WOTC will be releasing free, monthly supplemental game materials in it's Unearthed Arcana series.
The first of which just released this week. It includes some races (warforged, shifter, changling), optional rules (Action points based on the Hero points rules option put forward in the DMG) as well as a whole new class path (Wizard: Artificer). Not all of it has been playtested and you may need to tweak it for balance, but it is more free material.