We acquired the D&D Starter Set and Players Handbook and have begun playing the supplied adventure. Now I'm inspired to try my hand at creating my own adventure. I'm planning to pick up the Dungeon Masters Guide to have more information on how to do so, but between the DMG and the Starter Set rules will I have a wide variety of monsters to use? Or am I also required to get the Monster Manual for that purpose?
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The Dungeon Master's Guide (DMG) does not contain any monsters. It does provide advice on how to create your own monsters, which is useful for creating your own adventures, but it's a lot of work to create a single monster, let alone all of the ones you require for a full adventure.
However, the Monster Manual (MM) is not the only source of monsters. The DM Basic Rules are available for free online and contain a large number of the monsters from the MM, albeit without the extensive lore that accompanies them there. The same goes for the newly released SRD.
There are also limited lists of monsters in the free online supplements for Hoard of the Dragon Queen, Rise of Tiamat, and Princes of the Apocalypse. Finally, as I'm sure you're aware, the PHB contains a highly limited list of monsters, mostly low level, mostly beasts, that could be useful to you. There's nothing in there that isn't in the DM Basic Rules, though.
More than the Basic Rules list of monsters, Wizards has released the 5e System Reference Document that has 200+ monsters encompassing nearly all of the classic D&D list of monsters. It is found from page 257 to 360 of the 5e SRD PDF.
You might well create very engaging adventures using only character races (men, elves, dwarves, etc.) and natural animals (for which you could improvise stats, if you even need them -- riding horses and farm animals don't really need them, most of the time). Even if you want monsters, you could create your own stats for creatures out of various myths (that's how many of the first Monster Manual monsters were created).
That said, as noted in another answer, it's relatively a lot of work to create a monster that might well only be used for a single encounter (one of the reasons D&D has so many monsters is to keep the players from just automatically knowing how to kill anything they might encounter -- as used to happen with the limited number of adversaries in the original game of the 1970s). Using available sources is better use of GM prep time, unless you specifically want to create a campaign that doesn't use any of the existing monsters, for some reason.