I'll recommend Fidelity Brokerage for its wide variety of investing tools, for its complete tax reporting (i. e. , complete with cost basis -- so a simple import into Turbo Tax pretty much finishes your Schedule D work) in tax season, for its accuracy, and for a platform that is usually available and responsive. But, that recommendation comes with a caveat: There is no single best stock broker.
If you're a day-trading type, you might prefer a site with dirt-cheap commissions (which, alas, often are coupled with some drawbacks). If you're using software that permits you to enter trades, you may elect to restrict yourself to the subset of brokers who support trading from within that software product (e. g. , Investools, StockFinder, Omnitrader, and VectorVest are either currently permitting trading from within their software or have announced plans to support direct trading "soon". In all cases, only a subset of brokers -- and not necessarily the same ones -- will support the feature).