Should I resign five days after starting a job after learning of a better career opportunity?

You're still on probation and 3 weeks isn't enough time to be doing anything meaningful in the long run. You'll still need to be replaced, but now they've lost three weeks training you.

I suggest you don't overthink it and go with your studies. There's no ethical dilemna here.

Should I resign as early as possible? Also, do I have to go into detail about my graduate study with my employer, or should I keep things vague?

If you are 100% sure that you want to leave, you should resign tomorrow. There is no benefit to the company to having you around a few more days.

It would be professional for you to explain what happened to change your decision. Just explain it briefly, thank them for the opportunity, and be on your way. They won't be interested in a deep dive regarding what you will be doing in your studies.

It's unfortunate you accepted a job while having a better offer still outstanding, but what's done is done.

If this is a US government position: They may be forced to suddenly lay off some people anyway. So talk to someone high enough -- if they need to lay off ten people you might be one of the ten and save someone’s job.

Yes. If you've made the decision, and you really have the other position confirmed, then you should inform and pivot without delay. Rip the band-aid off.

Just be candid about the circumstances.

You applied several months ago to a highly competitive academic research position, and had really not expected that application to come to fruition.

In the meantime, you'd applied for and succeeded in your application for your current employment.

But the academic application has in fact now succeeded too, and you wish to take that opportunity.

A government employer overall is very unlikely to be left smarting about such a decision, at least if they still have personnel functions with the slightest perspective on what is the common good of public administration. The government wants people doing the academic research they also fund, and the small wasted costs here are just the costs of running public administration as a whole and of resolving people into their appropriate roles and occupations.

Usually public administration also works at a scale and pace where a young graduate who is a new hire would not be sorely missed.

Unless there is concern about lost wages or financial security, I would also advise notifying your current employer immediately, as at this stage they may be able to immediately offer your position to similarly competitive candidates who were also interviewed at the same time you were - whereas if you leave it weeks or months, those candidates may be lost, and the recruitment process will have to restart at an even earlier stage.

You always have to think of yourself. If you really want to go the academic way, then go for it.

Employers, especially when you have just started, dont owe you anything. If they had a change in circumstances, they would let you go in a heartbeat.

Its true that they have invested some time in training you, but they are a Government department. Unless this is a magical one, they havent really lost anything.

The more you delay, the more time they are losing. Just go and tell your manager or HR, what you want to do.

Ask your manager.

Is there something that you could do in the next week or two that would make up for or ameliorate that you're not staying? We don't know. Your manager should.

Maybe it's best for them if tomorrow is your last day. Or even this past Friday (presuming that's the last day that you worked). Maybe they want you to work a month's notice (a month from tomorrow), based on what your employee handbook says (from the comments on the question).

There is no reason to delay telling your manager and starting the resignation process. Worst case, they send you home immediately, which is what you want. The other worst case is that they make you work for another month from tomorrow. Not ideal but not horrible either. There is no real reason to work longer than the defined notice period. You mention no reason why you would want to continue to work. Telling your manager has no downside and may help keep you from burning that bridge behind you.

Note that if you wait three weeks, then your employer would reasonably expect you to work for a month notice period after that. That's more than seven weeks until you can leave. If you talk tomorrow and your employer wants you to stay longer than a month, they can of course ask. It's up to you whether you do it. If you want to be at graduate school as soon as possible, then the longer than standard notice period should reinforce your reasons for resigning tomorrow.

There are times when you should delay telling your employer. For example, if you need to confirm the academic offer or are waiting for the official, signed offer. However, you don't mention anything like that. You don't mention anything that isn't consistent with it being fine by you to not get paid for more than the five days you've worked. Why delay?

You can still work seven more weeks even if you resign tomorrow if that's what your employer wants and you want to agree. Let them choose. There is no reason to make that decision for them. Resigning after four weeks is not noticeably better for them than resigning after one week. Not taking the offer at all would have been better but is no longer an option. It's more likely that resigning immediately is better for them than a longer notice period that just wastes more of their resources.

You should be willing to work out your notice period. If they want that work, they're reasonably entitled to it. They aren't entitled to more than that.

Note: your graduate school offer apparently says as soon as possible. It's not possible for less than a month (your official notice period). Tell your manager tomorrow that you're resigning. Agree on an end date. Notify the graduate school that based on your required notice period, you won't be fully available until after that date.

It's barely possible that the graduate school may have room not to accept that, even though they waited pretty late to accept you. You might want to talk to them before resigning to verify that. However, there seems no reason you can't call them and still resign by end-of-day tomorrow if they say that's fine.

Ask your manager. (I am purposefully re-iterating the words from the start of mdfst13's answer.)

Keep in mind that regardless of when you leave, this experience might or might not blacklist you from the same government organization in the future. Such information might or might not end up in some records that get distributed where others may see this.

But, presuming you do wish to go forward with this, don't jeopardize your new plan by delaying too long. If you do feel like you can delay a bit to help your current government job, then perhaps be willing to do so. But do talk to the manager right away.

The quickest that I've ever had a job where I felt like they were done training me, and I was providing real value to them, was at a car wash where I only needed minimal training and was helping them within an hour. On the longer side, at some IT jobs or education positions, it could be months before it felt like new staff had enough familiarity with how the organization works that they could be pretty useful, and worth the significant training investment that has been expended.

In the mean time, there is additional resources showing what needs to happen, additional expectation for supervisory staff to be paying attention to how well things are going, additional stress even by co-workers who want to invest in you by having positive interactions and welcoming you to the team, and additional encounters with "customers" (whomever those are) who may may receive somewhat reduced quality in experiences as they are being helped by a trainee. All these unfortunate costs may continue to get invested in the hope that the investments will continue to work out.

If you let a manager know immediately, they might decide the best thing is to cease having some of those costs. Or maybe to utilize you a bit to help handle a back-pile of overdue work, but can then know to involve you in just simpler things and not those which will realistically need even more training investment, and maybe expose you to less confidential resources.

Let the manager know immediately (at least, once you arrive at your next shift). If the manager is not available to speak to, perhaps leave a message (send an E-Mail, let a secretary know, send a text message, etc.) Your message can be very brief: "I unexpectedly found out I will need to be exiting this position fairly soon. Please discuss further with me when available." Then, the manager can decide whether that person feels like it is more beneficial to cram a meeting in with you in the morning, later in the afternoon, sometime the next day, or whenever works best with that person's schedule. But at least then, any delay is on them, not on you.

Hey, congratulations on getting that job you were wanting when you applied to it. And extra congraluations on getting success with that other potential opportunity you applied to even earlier!