In context, when a Roman army marched on Rome, that meant blood would be spilled if they were opposed. This precedent was set first by Sulla, and it being only a few decades later the memory of Sulla's bloodshed would still be relatively raw. That Caesar had made the decision to bring troops into Italy, outside his province, meant that he meant business: such an act was illegal and would make him liable to charges of treason. In fact, part of the reason Caesar did this was to avoid charges of treason for the gross overreach of his imperium in prosecuting an eight-year war in Transalpine Gaul with troops he had not been granted by the Senate, when he had been granted Cisalpine Gaul only as a province. For Caesar to have taken this action meant either surrender on the part of his domestic opponents, or civil war. There was no middle ground at this point. War was all-but-inevitable, Caesar and his elite opponents would not stand for anything less.

Pompey was, judging by his stellar career, an excellent general, certainly on the same playing field of even Caesar. That he lost at Pharsalus has naturally exposed him to much criticism, including suggestions that he merely did the Senate's bidding. But considering his history and his position, I think it can only have been the opposite. Pompey was a dictator in all but name, and he did not have to bow and scrape to please the Senate. It was he they appealed to for a military response to Caesar, and it was he they gave the supreme command to.

From a military perspective, (it is probably more complex than this, but...) I think Plutarch's account gives ample reason for Pompey to have fled Rome. Caesar had thousands of hardened troops under him, already under the standards, and Pompey did not have an equivalent force. If he did, he would have fought. He went to Greece because it was there he was able to gather his own veterans, tens of thousands in number, whereas his veterans in Spain and Italy could not help him due to Caesar's dispositions. However, an army, even one of veterans, takes time to train and whip back into shape. This is the time he needed if he were to carry on a war with Caesar, who had at least ten legions under his command. When Pompey eventually faced off with Caesar in Greece, the former had greater numbers and the latter was suffering logistical difficulties. That Pompey lost the battle was by no means a foregone conclusion, and I cannot easily explain the result myself. The result appears to have been a product of a decisive flanking manouvere by Caesar's reserves (Lendon 1999 280, 320-2). Considering Caesar was fresh from his intense and complex Gallic campaigns and Pompey had not commanded troops in a long time, that Caesar was a bit 'sharper' on the battlefield should not be surprising.

Caesar's personal commentaries, the Bellum Civile, are a contemporary account of this war. I wish I had my copy on me (I'm not at home), and if I did I would have incorporated it into my answer. Goldsworthy is a decent writer and a good academic so his "Caesar: Life of a Colossus" is no doubt a decent place to start on this matter. Goldsworthy will provide a detailed reading list on any issue you are interested in during this period, so I'd probably get a copy of that book.

What is Plutarch's "Fall of the Roman Republic," is that some particular edition of the Lives that only includes the late Republican figures?

Caesar's advance into Italy was quite shocking. He moved unbelievably fast, so that Cicero in his letters during the period is not totally aware day by day of where Caesar actually is. For example, Ad Atticum 7.11 (January 19, two days after Pompey left the city) begins:

quaeso, quid est hoc? aut quid agitur? mihi enim tenebrae sunt. 'Cingulum' inquit 'nos tenemus, Anconem amisimus; Labienus discessit a Caesare.'

I ask you, what the hell is going on? For I'm in the dark here. Someone says, 'We hold Cingulum, we've lost Ancona, Labienus has deserted Caesar.'

It gives an idea of the confusion of the situation.

Ernst Badian said (at least in private, I dunno if he every used the phrase in publication) that the best thing that Pompey ever did for Rome was leave. Besides avoiding the devastation that would have been brought to Italy if he had decided to wage war there (which is evident from what happened during Octavian's wars) Pompey's flight to the south, and then from Italy entirely (which did not occur immediately--Pompey didn't leave until late February, and he left pickets at Brundisium to slow Caesar down and prevent him from crossing) made good strategic sense. Pompey had a huge base of support in the south--and indeed, Cicero mentions that when Pompey left the city and moved south the towns became more opposed to the idea of conceding anything to Caesar and rallied more strongly with Pompey. Further, Caesar's advance was so fast that a position in the city was essentially untenable--in less than a week Caesar overran Etruria and Umbria, and Pompey quit the city under Caesar's nose. The capabilities of the Pompeians in Italy can be guessed at by examining the few instances where open fighting broke out in Italy. By late January the Tenth Legion had reached Caesar's advance troops, with more further north, and with two legions he overran Umbria. Cingulum, which Cicero mentions above, despite being a foundation of Lentulus', deserted to Caesar, who advanced on Asculum, which was being held by Lentulus Spinther with ten cohorts. Spinther, rather than trying to fight against the overwhelming force that Caesar had brought, fell back--it's notable that Spinther's troops suffered heavily from desertion during the retreat. He turned what was left of his troops over to Vibullius Rufus, who was gathering Pompeian levies in the south--Vibullius also gathered up a couple more cohorts from garrisons fleeing south, and made his way to Corfinum, where Domitius Ahenobarbus was preparing for a siege with twenty cohorts. Corfinum is about due east of Rome on the other side of the Appenines, and by this time Pompey had already fallen back to the south--Domitius requested that Pompey relieve him, but the support never materialized and the city surrendered to the Caesarians.

Why Pompey decided to retreat to Brundisium rather than relieve Domitius, which Cicero thought was cowardly, is unclear, but the chaotic nature of the first few months should certainly not be underestimated. We get the idea from Cicero of a Roman aristocracy that has no idea what the hell is going on, and by the time people were ready for action it was often too late. In any case, Pompey's position in the east was extremely strong, one way or another. Pompey had been for several years the virtual ruler of a private empire, until the passage of his Eastern Settlement finally transferred his conquered territory to provincial control. Even in 49 he had clients, support, and funds in abundance in that part of the world, and was confident that he could muster a force vastly in excess of Caesar's, though some of those troops were destroyed by Caesar's lightning campaign into Spain. Further, the Pompeians had total control over the sea. To cross over, it was expected, Caesar would have to manage to build up enough of a presence in the Adriatic to transport his troops over, which would not be easy. Besides, there were Pompeians in Spain to his rear--Caesar ended up cleaning up that force remarkably quickly before turning his attention back to the problem of getting across the Adriatic. Pompey could, if he wished, simply continue to strengthen his forces while waiting for the right moment to cross back into Italy--every moment that decisive battle was delayed Caesar grew weaker and Pompey grew stronger. As it was, Caesar only just barely managed to slip his forces across the Adriatic, due mostly to good fortune, and up until Pharsalus he was in a precarious position, outnumbered and without supply lines. Nobody, not even Caesar, expected things to turn out so well for the Caesarians in Macedonia--the later portion of the de Bello Civili often reads as if Caesar himself is surprised