That link must be a few years old, some of the companies they rate no longer exist (including their “winner”, Datek, which has been subsumed by Ameritrade).
Any of the big on-line brokers are probably OK. You might also want to check your bank and/or credit union; some of them have investment partnerships now. Check specifically for the broker fee on a trade (somewhere in the $10-20 range is reasonable), and whether it’s higher for “limit” orders (it shouldn’t be more than $5 higher).
When you’re first starting out, you’ll mostly want to place limit orders (they “limit” the amount you’re willing to pay or sell shares for), since they cover a multitude of potential newbie mistakes.
Biggest advice, assuming you can afford it: Take a couple hundred dollars and just do it. Assume you’re paying the $200 as “lessons”, with the upside that you probably won’t lose the money (some will get eaten up in commissions). Trading isn’t that hard, but it’s scary to a lot of folks: every person I know who starts wishes they’d done it earlier.
It used to be that you generally had to buy/sell 100 shares at a time. In these days of computerized trading, you can buy/sell individual shares if you like. Remember that the trade commission is independent of number of shares, though:
Let’s say you want to by “FOOB” company (I made that up, if it exists, assume it doesn’t), currently trading at $10/share. If your broker charges a $15 commission, it will cost you $15 to buy the stock, and another $15 when you (eventually) sell it, for a total of $30 in commissions. Let’s say FOOB goes up to $12/share and you decide to sell:
So, if you buy 1 share, then it costs you $25 to buy ($10 for the share + $15 for the commission). When you sell it, you get -$3 ($12 for the share minus a $15 commission). Your stock market experience has cost you $28, which you can write off as a capital loss on your taxes.
On the other hand, let’s assume you buy a mere 10 shares. This costs you $115 ($10/share * 10 + $15 commission) when you buy. When you sell, you get $105 ($12/share * 10 = $120 - $15 commission). Now your loss is only $10.
If you buy 100 shares, it costs you $1015, but when you sell you get $1185, for a profit of $160. You pay taxes on that $160 either at your base rate, capital gains rate (usually lower) depending on how long you held the stock.
So you can see that with moderately moving stocks, it’s better to buy larger amounts. However, if you hold the stocks long enough for them to significantly gain in value, even small purchases can be profitable.
Disclaimer: This is not financial advice. I am not licensed to give you the financial advice I am not giving you here.