Just a note on terminology: 'agent' generally refers to a recruited source or asset of an intelligence service, rather than a permanent employee, who would commonly be termed an 'intelligence officer'. The Stasi's primary 'agents' were the almost 200,000 civilian informants, known as inoffizieller Mitarbeiters ('unofficial collaborators'.) The permanent staff of the Ministry was much smaller; less than 100,000 people at its peak.

I'd heartily recommend Gary Bruce's book The Firm: The Inside Story of the Stasi; it's a good grassroots history of the organisation, focusing on the personnel and operations in two rather unremarkable Brandenburg district offices (Gransee and Perleberg.) It also has a very good section on Stasi recruitment, which I'm about to quote at some length...

So, the main complicating factor for your eager young would-be chekist would be that the Stasi didn't accept applications; they approached you, not vice versa. Per Bruce: "if an individual did approach the Stasi, he was likely to be rejected on suspicion of being a mole."

Fundamentally, the Stasi was looking for candidates with strong ideological bona fides; as the sword and shield of the Socialist Unity Party, it was vital that the Stasi's officers were true believers in the Marxist-Leninist project. Again, per Bruce:

Ideally, the Stasi desired a straight Communist trajectory from youth to adulthood—a working-class family with parents supportive of the regime, involvement and preferably leadership position in youth groups like the Free German Youth or the Society for Sport and Technology, the learning of a trade, and time spent as a worker.

That last point is important: the Stasi placed a high premium on candidates with blue-collar backgrounds. Most Stasi employees had vocational educations; few had attended university. In general, the Stasi was wary of white-collar recruits whose education it did not holistically control. Like the KGB, on which it modelled itself, the Stasi operated its own network of training facilities and education institutions, the most senior of which was the Juristische Hochschule (College of Law) in Potsdam.

The best pathway to recruitment by the Stasi was undoubtedly to be talent-spotted during your compulsory military service. In the late 1980s, around 65% of Stasi recruits came from three groups: the police (3%), the army (21%) and the Felix Dzerzhinsky Guards Regiment (40%), a standalone military force under direct Stasi control and responsible for guarding key government and Party installations. Opting to join the Dzerzhinsky Guards when you enlisted would significantly increase your chances of being recruited by the Stasi.

The other factor that would significantly help your prospects was good old fashioned nepotism: according to Bruce, "47% of Stasi personnel had a relative in the secret police." The logic here is pretty straightforward: if you've deemed the father sufficiently proletarian and patriotic to serve on the ideological front-lines, there's a reasonable chance that the son will also pass muster.