As other's have already mentioned, you can't have a 401k on your own, because a 401k plan is specifically one that is through an employer. The retirement account that you can set up on your own with an investment broker is called an IRA, available as both a traditional IRA and a Roth IRA.
Either one could be better, depending on the situation.
Reasons why a 401k could be better:
1. Most employers will match your contributions with some percentage, meaning they actually put money in your account. For example my employer matches at 75%, up to 8% of my income. So I put 8% of my paycheck in the 401k, they add 75% more, so if my 8% is $100 then they add $75 and my account grows by $175 because of my $100. If your employer offers a 401k with matching contributions ALWAYS contribute enough to get the full match, and this is far better than anything you can do with an IRA at an independent broker.
2. A 401k is automatic. You set up your withholding rate and your employer automatically contributes a portion of your paycheck every payday. So its not based on you remembering to go write your broker a check, or waiting until you feel like you have the money. You are truly paying yourself first. Although most brokers will help you set up automatic contributions that take a bit of money from your bank account each time you get paid (or you can have your bank automatically pay the broker).
Reasons why an IRA may be better:
1. With an IRA yo can choose your investment broker and pick (almost) any investment you want. Whereas with a 401k you are stuck with the broker your employer hires and with the investment funds they choose to offer. But keep in mind that having too many choices could complicate and confuse things, so be careful to pick a good broker who will help you pick good investments.
2. With an IRA you can always choose a Roth. You'll need to look up the differences between a "traditional IRA" vs a "Roth IRA" but in most cases for someone your age the Roth is the better choice. Tax law does allow the Roth designation within a 401k plan but your employer and their broker have to choose to make it an option for you. If your employer doesn't offer a Roth option in the 401k, then the IRA is better if you want the advantages of a Roth (but you should still take the matching funds in a traditional 401k first).
3. With an IRA your money is not tied up with an employer. If you leave your employer you don't lose your 401k, but it creates a financial tie to that employer even after you've left. You can roll the 401k into an IRA but you have to consciously decide to do that, and you might incur fees when selling your investments and buying new investments within a new IRA. But an IRA is independent of any employer, so you don't need to do anything if you change jobs.