I am a beginner and have been watching a few famous games and have a question about: The Game of the Century Donald Byrne vs Bobby Fischer 1956. Sorry if this is a really obvious question but why doesn't Byrne exchange Bishop with rook at F8 or take/pressure the Knight at C3

If 1.Bxf8, then 1...Bxf8, attacking the White Queen.  White could play 2.Qb3 or 2.Qc1, then Black playes 2...Nxd1.  If White then plays 3.Qxd1, then 3...Qb4+ wins the White Bishop.  If the king takes instead, with 3.Kxd1, then 3...Qxd4+ 4.Ke2 Bc5, threatening 5...Qxf2+.  Best for White is 1.Bxf8 Bxf8 2.Qb3 Nxd1 3.Qxb6 axb6 4.Kxd1 Ra4 5.Bb3 Rxd4+ and Black is up materially and with a better position.

It's not a bad question.  It's not at all obvious why taking either piece is bad and you can bet both players checked the lines where white takes the rook or knight first to make sure it worked/didn't work.

If he took the rook to win the exchange (Bxf8), like billwall said when Fischer recaptures with his bishop (Bxf8) the knight is off limits because if Qxc3 then Bb4 wins the queen -- meanwhile black will win back the exchange with Nxd1.  So is this an even trade?  Not really, white's king is still in the middle and black's pieces can jump right into the game while white's king side rook (for example) is still out of play.  Not a big deal for amateurs maybe, but you can bet this spells more material loss for pros (example below).

Taking the knight on c3 leads to a similar circumstance after black plays Re1 winning the bisohp.  So is this on an even exchange?  Again no because white hasn't finished development (he needs to castle) and all of black's pieces are putting pressure on white which again will spell more material loss (again shown below).

thanks for the responses  - very helpful

Well, Americans, with their customary Americentrism, may regard this as the "Game of the Century", but I wonder if Russians (or other strong chess playing nations) agree with this view? There were many more important/decisive games of chess than this one during the last century. A century which saw the peak careers of such players as Lasker, Capablanca, Alekhine, Botvinnik, Karpov and Kasparov, among many other greats.

Perhaps Russians - and others - have their own "Game(s) of the Century"?

Who decides these things anyway?? The English-speaking chess media?

As i say I am fairly new to chess and know very little about the game let alone the history. I just came across this game and that's how it was titled.

Your point is a valid one, but one would hope that the chess community already realises this? I'm pretty sure most  "___ of the century" type desscriptions tend to reflect more than just the quality of the object (indeed most often it reflects those who are compiling such lists, titles etc), whether its "the snooker match of the century", the "football match of the century", or the "painting of the century". Rarely is there agreement and its almost impossible to have object criteria so these things are always a reflection of of all sorts of things...but we knew that anyway, right?