Question: This is my first time DMing, and I already have the Player's Handbook. I'm wondering which core book I should get next: the Dungeon Master's Guide or the Monster Manual? I am creating a campaign for four players, and I already have some of it done, but I would like to have better sources for the rest of the campaign. Which one is more useful?

Answer: 
The DMG is much more important.
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First, it's worth noting that it's pretty important to have all three books.

That being said, the DMG contains lots of useful information for a new DM. While more experienced DMs might scoff at the advice inside, there are still a lot of useful tables, maps, and suggestions in the DMG that are good for getting off the ground. 

Additionally, the DMG contains the list of magic items, as well as guidelines for creating your own monsters and magic items. With the sample monsters in the PHB and the creation guidelines, you could create your own monsters fairly easily. 

A lot of Monster Manual is (legally) available online.
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The Open Gaming License legally makes a large list of monsters available for free. You can access this list at various places on the internet ([here](https://www.5thsrd.org/gamemaster_rules/monster_indexes/monsters_by_cr/), for example). The list is missing some classic monsters, like the beholder, but for practical purposes it doesn't really matter.

On the other hand, a lot of the DMG advice and creation guidelines are not legally available on the internet (though many magic items are). Thus, buying the DMG gets you a lot more stuff you couldn't get otherwise.
I'd recommend getting the DMG first. You can create your own monsters (stat up natural animals, they're likely to be adequate for first level characters), stat up NPCs from playable races in the PHB, and save the "monster" stuff for later adventures. You will, however, certainly want the additional rules in the DMG; the PHB presents only the player-facing subset, and isn't in itself enough to run a game even if only PC races are present on both sides.

For monsters, between the [Basic Rules DMG](http://media.wizards.com/2016/downloads/DND/DMBasicRulesV05.pdf) and the [SRD](http://media.wizards.com/2016/downloads/DND/SRD-OGL_V5.1.pdf), both available at WoTC for download, most (but not all) of the monsters used in D&D are available. That inventory of monsters is enough to run a long campaign. (Some of the product identity monsters like Beholders and Illithids are only in the Monster Manual).