This isn't intended to be a comprehensive list of all game dev tools ever, more a simple list of starting points that can get you started!
There are two main places that people can start. You can either look at game mods, or start from scratch, so let's start with ...
Mods and Modding
Start with something you already know and love ...
You can dive into mod projects, using a tool-set often released with a game that allows users to modify the original game. This is often a great entry level way to see if you really have some game design chops, and the patience for the work. I started out this way making mods for the old Mechcommander and Neverwinter Nights games many moons ago.
You can start out with something as simple as making a short fetch or kill quest in an RPG and going from there.
Maybe you can find one of your favorite games that has a toolset. That way you'll have some context and a passion for the material before you start out.
So what are some good games to dive into?
Minecraft - The game that was the entry point to gaming itself for so many, is also a great place to start when it comes to modding. Given the games massive reach and popularity there are a virtual treasure trove of modding resources available.
Skyrim - The Bethesda Elder Scrolls series has a history of strong mod options and Skyrim is no different. It has an expansive and fairly fully featured set of mod tools that come bundled with the game. It also has a fairly vibrant community. You can check out places like Nexus Mods, the Skyrim modders reddit, or any number of great You Tube tutorials. If you prefer your RPG to be post apocalypse instead of fantasy check out Fallout 4's similar tools.
Super Mario Maker - A console experience takes away much of the complication and allows you to focus on just creating a fun experience for others players. While it might not teach you as many technical skills, it is a fantastic opportunity to focus on how you pace for fun in games.
Starcraft 2 - The editor that can be accessed for Starcraft 2 is pretty much the same one that the developers in-house use to make the game itself. It is an incredibly powerful tool that allows users to make all kinds of different game experiences. The Starcraft Arcade has an active development community that is full of support and advice. A good place to start is the mod tutorial forum or any one of the You Tube tutorials that exist, or a site like SC2mapster.
Divinity Original Sin - This throwback RPG is another robust editor that can add new content to the game with relative ease once you get your head around the interface. The developers have some videos online that show a lot of the functionality, and there is an active community on the developers forums.
Shadowrun Returns - Another old school RPG that has a bundled toolset that allows you to get to grips with creating content relatively quickly. There is a developer wiki, and you can find some You Tube tutorials
Civilization V - If strategy is more your thing there are some tools available for modding Civilization V that you can quickly sink your teeth into! A good place to start is the fanatics forum for modding.
Counter Strike: GO - A place many fledgling level designers have started out. The community is still going strong and has a near insatiable appetite for new maps. You can start with the developers wiki or hit up You Tube for some tutorials.
Source SDK - Counter Strike uses the Source tools, as with the above you need Valve's Steam platform for this, but it isn't only the venerable old shooter that you can mod. Most of Valve's modern games have mod tools built in, and all can be accessed from the Source SDK. While it doesn't have the most user friendly of introductions, it does have one of the most active and helpful development communities to ask questions of.
Starting from scratch
Starting from scratch can obviously be a little more intimidating, but there are also some great resources out there that can help you get started for yourself! First let's look at some of the tool-sets that exist out there.
Twine - Want to start with just some text? Twine is a text based game creation system that lets you create interactive fiction.
Stencyl - Provides a way to make simple games without the need for any code. It offers a relatively intuitive drag and drop system that can get you up and running pretty quickly.
Game Maker - A great, flexible, starter tool. As with any toolset it can be overwhelming to start with a blank page, but thankfully there are some great tutorials out there. We will get to that in a bit!
RPG Maker - Makes a very specific type of game, but it does that pretty well! If you aspire to creating something JRPG inspired than this is a good option.
Swift Playgrounds - More a way to learn coding in a fun way than specifically how to make games, but it uses making games as a way of learning to code, so some double value going on here!
Unity - The go to creation platform for indie developers. Has a higher learning curve, but also a much higher ceiling. You can ultimately make professional offerings with Unity, and it has an extremely active development community.
Unreal 4 - The professional development suite has recently started offering free access to it's power. What might have been an intimidating professional tool in the past is becoming a much more approachable option now. The blueprint system in particular can help you get a game up and running quickly. Check out their video tutorials for a head start.
Amazon Lumberyard - The other professional solution that isn't as intimidating as you might think. It offers a powerful set of tools, and some fairly robust example games and tutorials to get you going!
Almost all of these platforms have tutorials that will get you started with a demo game or two. Simply downloading them and working through the tutorial is a great way to start out. It will let you start to understand how things work, and give you insight into whether you enjoy working with game creation systems.