1. For basic advice, google the JD Collins stock series (sorry, I don't have the link).
2. Fidelity, Vanguard, etc. are brokerages. They buy and sell stocks, bonds, etc., and package them all together into mutual funds and ETFs, which they will then sell you shares in; many of them also offer an individual brokerage option in which they will sell you stocks and bonds directly rather than as part of a mutual fund, but they will charge you trading fees to do that.
A 401(k)/IRA/etc/ is more like a bucket in which you hold your investments that comes with certain tax advantages. For your IRA, you can set one up with any brokerage firm you want; for your 401(k), you are stuck with the brokerage and funds that your employer chooses. Fidelity/Vanguard and the like do run some people's 401(k)s, but not mine -- mine is run by a specialty shop that makes its money from managing companies' 401(k)s; they don't offer their own mutual funds, but they select the funds from other brokerages that we can invest in.
So say your 401(k) is with Vanguard, and you just want to buy indexes for simplicity. So first, you put money into your 401(k) -- you choose VTSAX. But then you also want to open an IRA. So you call Vanguard and open a separate IRA account, and you you also put that money in VTSAX. And now you have extra money you want to invest, so now you open a separate brokerage account with Vanguard, and you also put that money in VTSAX. Now you have three accounts with the same company, and they all own the exact same thing! One brokerage, three buckets, one investment. [You could also set up your IRA and post-tax account with Fidelity or anyone else; I just chose Vanguard for illustration]
Why this complexity? Because the different buckets have different rules.
-- The 401(k) is governed by your company's rules and the IRS rules. You can put in up to $18,000 this year, plus any employer match or profit-sharing. You cannot withdraw from the 401(k) until you leave the company, but you can take a loan if your company's plan allows it. You can change your investments, but only among the ones that your plan offers -- you can't send your money from Vanguard to Fidelity, for ex., or buy Fidelity funds if your plan doesn't offer them.
-- The IRA is governed by the US tax rules. If you make too much money and have a 401(k) at all, you can't open an IRA. You can only put in $5500/yr, but that money grows tax-free until you retire. You can't take a loan, but you can withdraw money if you need it -- but then you pay taxes and a 10% penalty if you're not old enough. You can also roll it over into a Roth IRA; there is no 10% penalty for that, but you need to pay taxes on the growth, and then you need to leave the Roth alone for 5 years before you can use that money. You can also buy and sell within the IRA -- e.g., now you want bonds, so you sell some VTSAX to buy a Vanguard Bond index -- and as long as you keep the funds within the IRA, there are no tax consequences. You can even transfer it to another firm if you want to (which you can't with the 401(k). [There is also a Roth option with its own rules, but I am ignoring that for now]
-- Note that those first two options give you significant tax benefits, but those benefits come with restrictions on how and when you can access the money if you need it. That is why many people keep some investments in a post-tax brokerage account (account #3). You pay taxes on any dividends/capital gains Vanguard pays out every year, and on any other growth when you sell it -- but you can sell at any time, for any reason. So you can sell this index to buy bonds, or to buy a house, or to pay for a kid's college, or to send money to some other brokerage that your brother-in-law works at, or whatever -- but then you pay taxes on the growth every time you sell. The big difference here tends to be the tax-free growth: every year, almost every fund with throw off some capital gains and dividends, and you get to pay taxes on that money -- even if you just reinvest it in the account. And that means that you have a little less left invested to grow over time, because the taxes suck a little bit out every year. Whereas in your 401(k) or IRA, there are no taxes until you sell, and so all that extra $ remains there and continues to compound.
So the tl;dr here is that the different buckets matter because they have different rules, and those rules affect what you can do with your money and what taxes you pay on it.
3. "Diversification" means "the specific investments you put your money in," not "the company that holds your money." The Vanguard and Fidelity S&P 500 indexes are both going to hold fundamentally the same stocks: stocks of the companies on the S&P 500 index. So you get no diversification whatsoever by putting half your money into the Fidelity one and half your money into the Vanguard one. You can safely keep your money with a single brokerage like Fidelity or Vanguard*, and then just worry about splitting your actual investments among stocks, bonds, cash, etc. as you like.
*This is limited to the large, well-know brokerages like these guys. I would never put all my money in some small advisor's hands and let them trade stocks and such for me -- yeah, these guys are supposed to be protected and such, but there's too much chance of a Madoff, and I don't want to have to worry about where my money is if the guy retires or gets hit by a bus or goes out of business for any reason. But Vanguard and the like, that's not a worry.