When it comes to gym leaders, the process usually begins with a recommendation from either a current gym leader/Elite Four member or a high-ranking member of the Pokémon Association (the international governing body that oversees the leagues—and yes, the champion is typically one of these members by default). Why a candidate might receive a recommendation tends to vary pretty wildly. Some gym leaders elect their own kin (such as in the cases of Lavaridge, Violet, and Fuchsia), while others choose trainers who impress them (as in the case of Petalburg or Lumiose). In rare cases, the Association recommends winners of local tournaments (Viridian, Veilstone) or unofficial gym leaders who defeat local gym leaders (Saffron).
Regardless of the reason, once elected, the candidate is then subjected to a test administrated by Association officials in the form of a battle. Should the candidate pass (i.e., win), they’re given official gym status and placement in their regional league. (Side note: There are many, many unofficial gyms out there. Anyone can open a place, call it a gym, and make a badge to go with it, but non-sanctioned gyms don’t count towards league requirements and often don’t follow league rules. It’s therefore not a good idea to challenge these gyms.)
Once sanctioned, a gym leader may serve a number of functions in their community, depending on that community’s needs and the gym leader’s abilities. Many gym leaders serve as teachers and offer classes on pokémon care, battling, or related local customs to local trainers and trainers-to-be. Others support their town or city’s law enforcement and/or public safety offices, and most gym leaders serve key roles in their area’s tournaments and festivals. Ultimately, all gym leaders are public figures and role models, and as such, they are, in general, expected to act on their best behaviors to serve as examples to their communities.
Of course, this is in addition to other, personal duties. Many gym leaders hold second jobs, not so much because the Association doesn’t pay them enough—believe me, the majority of them get paid more than I do as an Association member—but more because it’s something to do when they’re not maintaining a gym. The Unovan gym leaders, for example, consist of a fine artist, a museum curator, three restaurateurs, a teacher, a pilot, a model, and the owner of Unova’s largest construction company. For another example, one Kantonian gym leader runs a perfume company, two others are researchers, one is in medical school, one is still technically active in the American military, and the Cerulean Sisters apparently fancy themselves as “water ballet performers” … not that I have anything to say about that last one. The point here is that although a gym leader’s primary duties consist of maintaining a gym, battling, and serving in their community, this isn’t as hefty a list of tasks as one would think, and gym leaders are free to do whatever they wish in their spare time (within reason, given the aforementioned note about being a public figure).
The Elite Four is very similar to the above, but the requirements, both in becoming one and in performing as one, are more. For starters, it’s not possible to be recommended by a gym leader for a position in the Elite Four; candidates can only be chosen and recommended by high-ranking members of the Pokémon Association or Symposium. (To give you an idea of how high we’re talking about, I can’t recommend someone for membership into the Elite Four, but Steven Stone or Professor Kukui can.) The exam that follows is often harder than the one administered to gym leaders, and it often consists of battling four of the region’s topmost trainers (that is, the current Elite Four in most cases). And finally, while gym leaders are expected to serve their specific communities, the Elite Four is expected to serve their entire region. This means that if, for example, a threat to the entire region surfaces (to the scale of the Hoenn Incident a few years back), usually, the Elite Four is called into action to help the regional government with defense. Usually, anyway. As in, we don’t normally send single preteens out to battle the forces of nature, but for whatever reason, that’s been a trend as of late, which means the Elite Four has been called upon fewer and fewer times.
Besides that, Elite Four members are often asked to engage in regional events (such as the exhibition matches that typically take place before championship tournaments such as the Silver or Lumiose Conferences), teach new trainers whenever possible, and generally maintain their reputations as upstanding members of society. It’s just that as highly public figures, maintaining a certain public image is even more important to the Elite Four than it is to gym leaders, and oftentimes, it’s actually easier to be kicked out of the Elite Four due to violating the Association’s standards of conduct than it is to be kicked off the badge circuit. (I realize that sounds rather ghastly, but it’s important to keep in mind that the Elite Four are also the faces of their respective league. Any misconduct from them is a direct reflection on their region as a whole. It’s politics, in other words, only the leagues tend to take things a bit more seriously than actual politicians.)
Of course, as with gym leaders, the Elite Four (and champion, for that matter) can also opt to take second jobs, especially since they’re often called into action fewer times than gym leaders are. For example, it’s completely possible for Siebold of Kalos to run more than one high-ranked restaurant in Lumiose or Diantha to star in at least one feature film a year because, well, aside from the occasional challenger to the Elite Four gauntlet, the Kalos League rarely requires the services of its Elite Four.
In short, if you’re considering joining either group, anonymous, best of luck, but consider beginning with the gym circuit first. Impressing a gym leader enough to get a recommendation is far easier than impressing an Association member.