There are many different ways to save for retirement and strategies can vary a lot from person to person, but for most folks I would recommend either a traditional IRA or a Roth IRA.
If you are fortunate enough to have a 401K or a 403B plan at your work you should first maximize your contribution enough to take advantage of any employer matching plans that are available.
That is the first step you should take. Most employers who offer a 401K (and I will use that as a generic term to include 403B plans, since they are very closely related) have some sort of matching figure whether it is dollar for dollar up to 3% of your salary or matching 25% of your contribution up to a certain amount. This employer matching fund gives you an immediate return on your investment and you will not find anything like that anywhere else. For example, if you contribute 5% of your salary and you make $50,000 annually, your contribution will be $2500. This is a pre-tax contribution, so it also reduces the amount of tax you are paying to the government right away. This amount may also be matched by the company you work for. So if they have a dollar for dollar match to 3%, they will contribute $1,500 to your retirement fund. This is like getting an instant 60% return on your investment!
If you do nothing else, and I will clarify, you contribute nothing else to this account, but let it accumulate a 10% return in a mutual fund for 20 years, that $4,000 will grow to $27,500. If you continue to contribute at that same rate, it will grow to over $325,000 dollars. Most of that initial investment is from your contribution, but a great deal of it comes from your employer.
The next bit is a further savings. You can contribute after tax dollars to a Roth IRA and the money in that account will grow at a similar rate to your 401K. The difference between the two is that when you withdraw money from the Roth IRA it is tax free.
Now the question is, where is the money coming from for your investment? For the 401K that is a bit simpler than for the IRA. I recommend that whenever you get a raise at work, you contribute that to your 401K. You have learned to live on your salary (hopefully) and the raise going to your 401K is not going to be missed. For example, say you get a $2,000 annual raise on that $50,000 salary we used as an example. If you raise your 401-k contribution by 4% you will be adding $2000 to your 401K, reducing your tax base and adding another $175,000 to your retirement total! And that is only having one raise at year two. Imagine doing that every year or every other year.
On the IRA, I would recommend investing any windfall amounts and any tax returns you get. As of the writing of this article, you can contribute up to $4,000 per year into your account.