If you have access to an arcade stick (or are willing to buy one), I'd start with Super Street Fighter 2 Turbo. It's cheap, easy to pick up and play (assuming you're already familiar with Fireball and Dragon Punch motions), and introduces a lot of the archetypes that are in pretty much every fighting game, so you'll have a baseline level of skill at practically any other fighter. The guys over at Shoryuken also put together an Ebook about getting into Fighting Games, focusing on Street Fighter 2 that I think is pretty solid at conveying what you need to know to get the hang of it.
If you don't want to make that kind of monetary commitment to the genre just yet, I'd recommend giving P4A or a Marvel game a shot, because they use a 4-button control sceme rather than a 6-button setup, which makes them easy to learn on a gamepad. They're pretty simple, control-wise, and offer up a lot of the basics. My only complaint with Marvel is that it kind of goes bananas pretty quickly, so it's tough to get the hang of the basics if you're a beginner.
Also, the new Killer instinct's training mode is one of the best I've ever seen at introducing fighting game concepts. It'll teach you about spacing, combos, and even go into how the systems of a fighting game work. It's a great primer for the genre, and it's really easy to pick up and play.
I kind of like the idea of using a turn-based game like Fire Emblem Awakening to teach some of the core competencies of an RTS/MOBA at the player's pace.
One of the things I appreciate most about Awakening is that it surfaces all the data the player needs to make a decision when moving a unit, from enemy movement range to your win/loss percentage to detailed breakdowns of all the statistics involved.
Players can simulate a move, see a breakdown of its effects, and then simulate moves again and again to find the optimum result before committing to an action. It seems like a pretty good way to allow the player to experiment with different types of strategies in a safe space, and to show the amount of variables the player will need to keep in mind before putting it all into practice in real time.
There might be other games better suited to teaching some of those tactical skills up front, but Awakening seemed pretty beginner-friendly while still offering a whole lot of depth for people who really wanted to dig in.