Casca is the first to strike, and, after each of the conspirators attack Caesar, Brutus is the last to stab him. The script of the play does not indicate any other order, except that all of the conspirators do stab him at least once.
In Act III Scene 1, all we have to go by to answer your question is the line “Speak hands for me!” from Casca, followed immediately by the stage direction "They stab Caesar." Simple enough, and hardly detailed. We can only assume that Casca was the initial stabber, his spoken line being an indication that he got the onslaught rolling. And Caesar’s last words—“Et tu, Brutè?—Then fall, Caesar,” do not necessarily indicate that Brutus was last to lend his knife to the cause, but it is generally assumed that he was. Having Brutus stab last is a very symbolic gesture, Brutus being one of Caesar’s closest friends and confidantes; his betrayal strikes both Caesar and the audience all the more strongly when he comes last in line, and leaves the door open for Brutus's psychological anguish in later scenes.
As for the rest of the conspirators—and there are many—Cassius, Cinna, Trebonius, Decius Brutus, Cimber, Caius Ligarius—there are no specifications as to the order in which they fall upon their leader. And, indeed, it is hardly important, the deed itself being greater than the men who commit it.
According to one version from the Folger Shakespeare Library of the book Julius Caesar by William Shakespeare, in Act 3, Scene 1, just after line 84, Casca is the first to strike, then the others follow suite. In one movie version of the same thing, Brutus was the last, though you could tell he was completely unsure and really thinking if he should have planned and allowed it all to happen.