I'm looking into getting into D&D as a player. Is the free Basic Rules they supply on their website enough for a player to start, or should I just buy the Player's Handbook?
The basic rules are absolutely sufficient to get into D&D, try it out, play with it for a while and even run a campaign (though you may find that you want more options, which the PHB and Monster Manual provide).
First and foremost, if you're just looking to get together, build characters and run a home built adventure, there is enough information in Basic D&D to get you going. There are four races and four classes which give you 16 different PC combinations and each class has a few selections you can make within the class to give you a small number of more options.
The Basic DM book provides some basic magic items and monsters to get your adventure starting, there is a pretty solid list of monsters available from CR 0 all the way up to CRs in the late teens, which should provide plenty of challenging encounters from L1-20.
However, there are only 4 races and classes, which if you have more than 4 PCs may prove a bit lacking, or you may find that the options presented (each class has only one of the 3+ options each PHB class has). This would mean purchasing a PHB. You may also find that the list of monsters doesn't meet your adventure needs, in this case you'd need to either build your own monsters (DMG helpful, but not required if you're good at finding design patterns), or buy a monster manual for better variety.
Lastly, if you don't feel like writing your own adventures, you might consider getting one of the published adventures or seek out a free one on the web. There are tons of resources available, from the fairly expensive (though both long and fairly well produced) official ones, to much cheaper 3rd party ones (It's worth noting that there are monsters and magic items in the published adventures that are not available in Basic, for these Wizards has committed to publishing supplements that allow for the published adventures to be run with no additional cost).
So in summary, you can absolutely get going and play for a while with Basic D&D, but you may find that you feel somewhat constrained by the limited number of options in it and want to pick up a PHB/DMG/MM book. I've found all three to be quite helpful, but again, they aren't essential, at least not for starters.
The basic rules are enough to play with. They give you a basic subset of the game mechanics - you don't get a full list of races or classes (you get only the classic 4 classes, and only a selection of their versions), and you don't get any rules for feats, which are only optional to use anyway. The spell list is also not as large as the PHB, since several spells are only for classes not included in the basic rules.
Having said that, the PHB does include a number of races, classes and options that might enhance your play experience.
As the others have said, the basic rules are sufficient. One thing I would like to point out though: don't discount the ability for a person to freeform DM. If you get reasonably familiar with rules, you can run an adventure with just some dice and paper, compromising with your players as you go for the best experience. Just remember that you don't always have to follow the rules by the letter, so don't worry too much about getting the official rules. If you really want to make sure that you are observing every detail, then you'll want official rule books, but it sounds to me like basic rules will definitely suffice for you, at least for now.