Despite mecha anime (that is, shows involving giant robots) being one of the medium's oldest and most established subgenres, many newer fans find it difficult to penetrate the genre. Mecha houses many of anime's greatest classics, but a not-insignificant portion of fans out there view the very concept of giant robots punching each other as somewhat dated, or perhaps silly in nature.
It's certainly understandable that not everybody will be as thrilled as mecha fans are by animated robot battles, but luckily there's a wide range of material out there that involves giant robots, but does not feature those robots as the primary appeal of the show. Newcomers to anime shouldn't be turned off by the mere presence of mecha in a show, because there's a diverse range of material out there that subverts, sends up, or twists the typical robot tropes.
Neon Genesis Evangelion
The all-time classic Neon Genesis Evangelion is often described by fans as a "deconstruction" of the mecha genre, in that instead of telling a heroic robot story wherein the protagonist has to overcome their enemies on the path to self-actualization, Evangelion tells an experimental and introspective one that changes the characters irreversibly in the process.
The audience does get a good amount of robots punching monsters, but although that's the setup to the show, not much is quite what it seems. As the episodes go on, the show becomes less and less about the action scenes and more and more about the internal psychological drama of the characters.
Full Metal Panic!
The mechanically-equipped counterterror organization Mithril has identified Kaname Chidori as a critical strategic asset to future war efforts, the only problem being that she's still in high school and can't exactly ditch class to deal in matters of international terrorism. To account for this issue, Mithril deploys the no-nonsense pilot Sousuke Sagara to act as her undercover bodyguard.
Predictably, tasking an experienced robot pilot with posing as a student results in some mishaps that the show mines for comedy. Furthermore, Kaname's status as a high-value target for sinister forces invites danger to the school. Sousuke must at once learn to adapt to the unfamiliar world of high school life, while keeping his skills sharp enough to keep Kaname out of danger.
Martian Successor Nadesico
What would happen if a bunch of mecha fans were tasked with defending Earth, with robots? Nadesico has the answers, but there are also some unexpected occurrences along the way. The unlikely crew of the battleship Nadesico are the last line of defense for humanity, but they might not all be up to the task.
Nadesico at once plays into and subverts the classic tropes of the mecha genre; there are elements of parody in the outrageous hot-blooded protagonists, the show is also smart enough to regularly betray expectations by sending up those tropes at just the right time to keep things interesting. The show feels like a meta love letter to all things anime, as well as a competently done mecha action story.
Darling In The FranXX
A co-production between A-1 and Studio Trigger, Darling in the FranXX is an outlandish, Freudian, robot show set in a post-apocalyptic future where humanity has been driven behind the walls of domed cities by enormous, aggressive monsters. The mecha designs are extremely distinctive, and the action sequences are consistently high-energy.
Complimenting the robots, however, is a romance subplot that's absolutely soaked in melodrama, which also makes up the more memorable elements of the story. Feelings of romance are often too fickle or too committed in FranXX, which leads to the characters suffering as many wounds from love as they do from monstrous alien dinosaurs.
Turn A Gundam
By far the most laid-back of the mainline Gundam titles, Turn A deals primarily with the build-up to interplanetary conflict rather than starting in medias res like most of its counterparts do. The result is a melancholic story about characters trying to stop a war from happening, rather than winning it.
One of the critical differences between Turn A and other Gundam entries is the setting -- in Turn A's far future, humanity's technological prowess has regressed to that of shortly after the industrial revolution, so the robots encountered in this setting are relics of a more advanced time. This, combined with the relative lack of action, give Turn A a unique atmosphere that sets it apart from other entries in the franchise, and mecha anime in general.
Code Geass: Lelouch Of The Rebellion
With the Empire of Britannia threatening to establish total military and political dominance over the globe, one man named Lelouch, equipped with a mysterious power, rises up in rebellion. Featuring constant scheming, outrageous plot twists, and constant betrayal and backstabbing, Code Geass is a thriller that isn't to be missed.
Although the show features a healthy dose of mecha action scenes, the real meat of the show is the intrigue and plotting found in the many twists and turns. It's tons of fun trying to keep up with the plans of these characters and to speculate about what their real motivations might be, because it's almost never what the audience expects.
Bokurano
Fifteen children are tasked with protecting the Earth from fifteen consecutive alien invasions with the help of an enormous robot of mysterious powers. It's a classic setup to a mecha anime, but one that's quickly subverted when the story takes a much darker turn -- the power source of the show's robot is more sinister than it initially appears.
Robot fights are very much not the focus of the story here; instead, the audience spends most of their time getting to know the backstories and struggles of the fifteen characters that make up the show's main cast. Without venturing too far into spoiler territory, this anime veers very rapidly into tragic territory, so don't go in expecting an exciting a fun robot action story.
Eureka Seven
Robots that fight with surfboards in the sky should be enough to sell anybody on anything, but those who are still skeptical can rest assured that Eureka Seven, despite its incredible animation and mecha fight choreography, is mostly a love story with elements of robot action thrown into the mix.
Indeed, the romantic leads and supporting cast get most of the attention here, and although the robots are an important element of the plot, there are many episodes where viewers never see a mech leave the flight deck. In addition to the central romance, there are a number of well-timed plot twists about the characters and setting that keep things from becoming stale.
Patlabor: The Mobile Police
Patlabor imagines a world where running into giant robots is as ordinary as seeing a construction crew or cop car while driving to work. Originally conceived as construction tools, labors (the show's term for giant robots) also gave rise to a new type of crime: that involving said giant robots. To combat this trend, the police of the setting are equipped with robots of their own.
The misfits of Tokyo P.D.'s Second Special Vehicles Division encounter all sorts of hijinks while driving labors around the city; in a world with giant mechs all over the place, traffic disputes, DUIs, and domestic disturbances can become escalated affairs that end in mecha battles.
Terrestrial Defense Corp. Dai-Guard
An alien race thought defeated returns to destroy mankind, and the last line of defense between them and Earth is... a bunch of office workers? With the military moving on to bigger and better things, it's up to a handful of lowly office staffers to restore the now obsolete Dai-Guard, a very old giant robot, to working order and battle the alien menace.
Battling bureaucracy and budget limitations more often than aliens, the woefully unqualified cast of Dai-Guard have to stumble their way through anime's most hilariously mundane mecha action show. As it turns out, dealing with giant robots isn't quite as romantic and exciting as it's cracked up to be.