Should I contribute to a 401k pre-tax or Roth after tax if I plan to work for a year and then go back to grad school?

First and foremost, any money you put in, you won't lose. Any money your company puts in, you will have to check what the vesting structure is. Typical examples are: 100% at 3 years (which means unless you work for 3 full years, the company gets the money back) or 20% every year for 5 years (which means at 1 year anniversary you now get 20% of what the company has put in, at 2 year anniversary 40%, up to at your 5 year anniversary, you get 100%).

Secondly, the terminology for 'match up to 100% of 3%' means that you can put in up to 3% of your salary, and if you do so, the company will match that dollar for dollar. If you put in less than 3%, the company will only match up to the amount you set aside. If you put in more than 3%, good for you for saving more, but the company won't be putting in any more.

Except in the most extreme circumstances, the advice here is to always contribute enough to get as full of a match from your employer as you can. Because the return is otherwise unbeatable. I.e. 100% match == your dollar immediately becomes two dollars, and you are not going to find a 100% return on anything outside of maybe a lottery ticket.

Now, combining those two answers above with your plan to return to grad school -- you may have one of those most extreme circumstances. That is, if you know for sure you are leaving before your employer contributions will vest, then maybe it is not worth contributing.

That said, there are a lot of people who were 'for sure' leaving their jobs back in, say Jan of this year, who are still working at that same company because pandemic came and jibbered up their plans and don't want to risk quitting now and finding a new job in this current market. I am not here to toss cold water on your grad school plans, but life changes. The school you want to do to may shut that program down. You may decide that this working life ain't so bad. Etc. Etc.

In short, even if you don't stay long enough to get the vesting, I'd most strongly recommend still contributing enough to get the full company match. Because if you do stay long enough and haven't been contributing, I think the regret from not getting that match will be great. And, as above, what you put in still stays. And getting into the habit of having savings taken from every paycheck is by far the easiest way to establish good savings habits. I.e. you don't learn to spend what you effectively don't have.

The Roth vs. traditional is not the easiest question to answer. If one's situation were to remain static over their entire life, then it would not matter. I.e. if one made the exact same salary, had the exact same deductions, and had the exact same tax brackets their entire life -- it does not matter if one pays taxes now (Roth) or later (traditional).

That said, the above is true for nobody, real life earning, deductions, tax brackets and so are always moving.

A reasonably decent assumption here is that since you plan to go to grad school, your future paychecks will probably be larger than your current ones, and hence you are likely to be in a higher tax bracket later. In this case, it makes a little more sense to do Roth now. I.e. the taxes you are going to pay now to put money in are likely to be relatively lower than the taxes would pay out later to withdraw. However, this is not a given, depending upon what your retirement plans are, and quite simply whatever Congress does in the future.

The ultimate hedge here would be to see if your plan can do both. I.e. if you choose to contribute say $200, can you put $100 into traditional and $100 into Roth, or whatever blend seems right to you.

Is it worth to invest on either the 401k or the Roth?

Yes (I'll discuss which you should invest in below)

Will I lose that money?

You won't lose what YOU contribute (other then the change in the value of whatever you invest in), but you may lose some or all of the matched funds if the company has a vesting period of more than 1 year. Most companies have a 2-4 year vesting period, so if you leave after one year you'll lose 50% for a 2-year period, 75% for a 4-year period (you'll get 25% for serving 1 of 4 years), etc.

what does it mean that the company matches 100% up to 3%?

That means the company will match 100% of what you contribute, up to 3% of your salary. If you make 5k per month and contribute 150, the company will also contribute 150. If you contribute more, the company won't match anything over 3% (but you still keep what you put in as per the answer above)

From the example, it would mean that if I make 60,000 I would be having 1,800 (my 3%) plus 1,800 matched by the company, for a total of 3,600 saved for retirement?

Correct.

The main factor for choosing between a after-tax (Roth) and pre-tax (Traditional) 401k is your tax bracket. If you are in a lower tax bracket now than you expect to be in retirement (remember that after-tax retirement fund withdrawals are "taxable income" for bracket purposes). then a Roth makes more sense. You pay a lower tax rate now in exchange rather then a higher tax rate at retirement.

If you plan on maxing out your contributions (or even the match), then a Roth makes more sense. Since the match is the same either way, you can effectively contribute more to the Roth than you can the traditional since you pay the tax up-front.

For example, if you put in 1,800 now and are in a 20% tax bracket, you will pay 360 in tax on that part of your income. You've paid 2,160 for a retirement balance of 1,800 that is now tax free. That's an after-tax return of -17.7% ((1,800 / 2,160) - 1)

If you put 1,800 in a traditional plan, you'll save 360 in tax now, but the fund will be taxable when you withdraw it. So you've paid 1,800 now for an account that has an after-tax value of 1,440. That's a "return" of -20% (your tax rate). So putting in the same amount after tax gives you a better starting point than putting it in pre-tax. (You could even out the math by investing the tax savings, but hardly anyone does that)

Note that this does NOT include the company's match, which is always pre-tax and adds 100% to your initial return. Since it's the same in either account type I left it out to make the comparison cleaner.

Of course, maxing out a Roth only makes sense if you can afford to put in 3% in a Roth pan without the tax deduction. You'll need to prioritize it appropriately in your budget.