Welcome to the wide world of D&D and RPGs in general. This is a common question from new GMs - where do the adventures come from and do I write my own or use pre-written ones?
Quick terminology check - "an adventure" is a single story arc that may last several "game sessions." A "campaign" is a longer arc that usually encompasses one set of characters going through multiple adventures, and a "campaign setting" is the game world you're using. An old term for a published adventure is a "module." A set of adventures designed to link together and carry you through a large part (or all) of a campaign is nowadays called an "Adventure Path" (previously: a "mega-adventure" or "module series").
Roll Your Own
Some people really take to coming up with their own adventures, and after the Starter Set equivalent (in earlier days it was the D&D Red Box) just make up their own adventures and even settings using the monsters and other bits provided. That's completely fine to do. You don't need the information coming in the core books to do it; there are many RPGs whose entire rules are smaller than just the D&D Starter set and people have created campaigns for them just fine. When those books come out, they'll have more rules and monsters and advice for you to use inside your own adventures if you want.
Modules, Modules, Modules
Other people like to use either partially or exclusively canned adventures. This has the advantage of using other people's work instead of having to do it yourself, and in some cases with published adventures that work is very good (and has maps and art and associated trinkets). The downside is that no published adventure is perfect for you, your group, and your set of characters out of the box, so you will have to tinker with it a little at least and may find it confining. This question shouldn't be answered with a list of the 5e adventures that exist, but there's already a bunch - ENWorld held a contest to generate some and various third party publishers are already hard at work coming out with adventures to keep you busy. On RPGNow there's both WotC and third party adventures coming available in PDF. People are frantically converting and creating adventures and most every blog is starting to be festooned with them, so if you want to use adventures others have written, there's already more than you can use.
Also, 5e has been made more "rules light" in the hope that it'll be easy for people to convert adventures from other D&D versions to it; if you feel confident enough in your cross-rules skillz then you should be able to use adventures you find online or in new/used bookstores for other editions fairly readily in your game.
Most people end up doing a mix - using some canned adventures, using parts of adventures, remixing adventures, intermixing them with their own content. There's no right answer, it's mainly about what you and your group have the most fun doing!