Round up of advantaged retirement investment approaches in the US?

Common tax-deferred vehicles include:

• 401k: an employee benefit offered by for-profit organizations, sometimes with a match
• 403b: an employee benefit offered by not-for-profit organizations
• SEP and SIMPLE IRAs: an employee benefit offered by for-profit organizations (often smaller organizations with a single owner)
• Traditional IRA: an individual's self-directed account held in trust by a financial institution.
• Roth IRA: an individual's self-directed account held in trust by a financial institution
• Buy and hold securities

The employee benefit plans are offered by the employer. The youth should ask the HR department if they are eligible. In the event they separate from the employer, they can stay in the plan as long as the employer lets them; and if not, or at their discretion, can create a "rollover IRA" (similar to a Traditional IRA) at a financial institution.

Traditional IRA and Roth IRA are set up at a financial institution by the youth. Traditional IRA generally reduces income tax in the year of contribution, and is taxed as income when distributed. After the age of 70.5, there is a required minimum distribution based on life expectancy.

Once the total income of the year is assessed, it is possible the contributions to a traditional IRA do not reduce tax in the year of contribution (it phases out for high income earners; and the limits are tighter if the taxpayer is eligible for a 401k or 403b from their employer). The contributions shouldn't be taxed when ultimately withdrawn, but this confounds bookkeeping until retirement age.

Roth IRA distributions are not taxed when taken in retirement age, or under certain pre-retirement scenarios (illness, first home purchase).

Buying and holding securities has certain advantages based on current tax law, though since the federal income tax was established in 1913 the rates and rules have both increased and decreased every few years. For several periods, capital gains have been treated as ordinary income. Advantages in the current rules compared to IRAs include exact control of the year of incurring the gains; step up in basis in estate tax; and capturing losses that currently can be used to offset some ordinary income. A disadvantage is in the event of a corporate merger, the owner may lose control of the year of taxation.

The rough heuristic for an 18 year old should be:

• contribute to a 401k with a match to capture the match percentage
• contribute to Roth to the limit
• contribute to 401k or traditional IRA to the limit
• buy and hold

Whether Roth or IRA are prioritized in the second step can change closer to retirement age and is a conditional function on the marginal income tax rate in the current year, the best guess of the marginal income tax rate in retirement, and at the point of forced RMD at age 70.5.

Letting a parent monitor an account. The youth can execute a power of attorney letting someone else monitor and control investments on their behalf. Well established financial institutions should have a form for this, for example.

South Korea and Social Security You didn't ask about Social Security directly, but did ask about automatic government plan participation. All regular US employees participate in Social Security with contributions automatically deducted by their employers. South Korea has a similar system, and there happens to be a reciprocal treaty between the nations where credits in one system can be used in the other for full eligibility, in the event the individual does not have enough in the place where they retire.

How do you mechanically do this (in the US?) Is it like a government function or something and the money goes through some gov't department?

That's the mandatory Social Security taxes, which you are not referring to.

Do you basically open a brokerage account and tick a box "401" (or one of those abbreviations)?

That's exactly how you do it, by opening a Roth or traditional IRA at a brokerage (Vanguard, Fidelity, Schwab, etc, etc).

Note, though, that there are two methods (the employer-sponsored 401(k), and the employee-driven IRA), and within those, there are two types (traditional pre-tax contributions, and Roth after-tax contributions).

• 401(k) programs usually have an "employer contribution" component which is typically 2% of income. Part-time employees might not be offered that, though.
• The IRS will eventually get it's slice of her flesh; it's up to her to decide whether to pay income tax now or later.
• For someone earning not too much money, pay taxes now because the tax rate is so darned low; possibly 0%.

Thus, I agree with Orange County: she should invest some savings in a Roth IRA (or possibly 401k if her employer offers it, and contributes extra); the money will not only grow tax-free, but future withdrawals will be tax free.

there is some chance that Youth ... will suddenly leave the USA, move to Korea or something and never return

EDIT: paragraph removed because it's not correct. https://www.schwab.com/ira/roth-ira/withdrawal-rules

For investing for retirement, there are four main types of accounts:

• Standard taxable account: There are no tax breaks with this one, but no restrictions, either. If you will need the money before retirement age, or you want to invest above the limits of the tax-advantaged accounts, this is your primary option. When you set up a standard investment account with a brokerage or a mutual fund, this is the default.
• Employer plan/401(k): This is an employer plan, so if your employer offers it, you can use it. Generally, anything you contribute is deducted from your income taxes, and you don't pay any tax on any growth until you take the money out at retirement. Usually, this account has only limited investment options. Sometimes your employer will match some of the money you put in. It is most often called a 401(k), but some employers will offer a 403(b) or a SIMPLE IRA, which are both very similar to a 401(k). Finally, some employers will offer a Roth 401(k), which is a 401(k) that is taxed like a Roth IRA (described below). To participate in an employer plan, all you need to do is tell your employer how much you want to contribute, and they will automatically take your contribution out of each paycheck for you.
• Traditional IRA: This you manage yourself; it does not go through your employer. Anything you contribute is deducted from your income taxes, and gains are not taxed until retirement age. You can invest in almost anything inside an IRA account. To set this up, you simply designate it as such when you set up your investment account. There are limits to how much you can contribute/deduct each year.
• Roth IRA: With this IRA, you don't deduct your contributions from your income taxes; you fund this with after-tax money. However, once funded, it grows completely tax free, and you don't pay any tax when you take it out at retirement age. In addition, whatever you put in, you can take out anytime if you need to.

If you have an employer plan available to you where your employer is matching some of your contributions, that is the best place to start. Your investment grows immediately as you put money in, due to the match. It is essentially extra money that your employer is giving you. Even a seemingly small match is worth it.

For a young person who wants to save for retirement and does not have an employer-matched plan available, a Roth IRA is what I would recommend. Because they currently have a low income/tax rate, they would not benefit much from the tax deduction of the 401(k) or the Traditional IRA; it costs them less to participate in the Roth than it does a high income individual. And once you pay the tax, you are done; the money grows in the account tax-free and no taxes are due when you take it out down the road.