IntelliJ IDEA was created and is maintained by JetBrains, the same company responsible for Kotlin. Because of this, you'll find no better IDE for working with Kotlin - the levels of integration and presence of useful tooling is unmatched.
Community Edition is completely free and contains everything you need to get started with Kotlin. There's also a paid-for Ultimate Edition (which is free for students), which contains many advanced tools and support for languages outside of the Java ecosystem as standard. Ultimate Edition is intended for use by professional developers, and may be overkill for new programmers.
For a quick guide on getting started with Kotlin in IDEA, see the Kotlin documentation.
For people that aren't a fan of IDEA, JetBrains recommends the Eclipse IDE. This is maintained by the Eclipse Foundation, and is completely free. It targets the Java ecosystem, and a Kotlin plugin is available for free on their marketplace.
For a quick guide on getting started with Kotlin in Eclipse, see the Kotlin documentation.
Gradle is a popular build tool, aiming to provide a rich API and ecosystem of useful plugins. While not necessary for beginner projects or small tasks, Gradle is an essential tool for working with larger, more complex projects.
Orchid is a documentation generator, able to generate a website from a set of Markdown files and documentation comments.
Orchid makes use of Dokka under the hood. Dokka is currently in quite a basic state, so documentation generated with it (or Orchid) may be a little lacking, but it's the best option we have at the moment.
KTLint is an anti-bikeshedding Kotlin linter with a built-in formatter. It's designed to help you keep your Kotlin code in check - from adhering to style guidelines to catching common mistakes.
Detekt is a static code analysis tool for Kotlin, available as a standalone tool or Gradle plugin, and with integration with IDEA. It provides more comprehensive code analysis than KTLint (and actually can be configured to include a KTLint analysis), and is an essential tool for people that wish to write top-notch Kotlin code.