Also, is there any recommended reading for Germanic/Celtic/Slavic, etc. migrations prior to the fall of the Roman Empire? Or also including during and after the fall of the Roman Empire, if there are no books that cover specifically before the fall.
/u/JimothyTheRooster provided a very satisfactory answer, but since this question is so close to my own interests and research, I'll add a few things:
What this tells us right away is that the Celts were far from an organized body - they probably more closely resembled a loose confederation of tribes who acted in concert only when they decided it would be to their benefit. The details of the divisions of these tribes are obscure and have been interpreted quite differently over the years. It is generally agreed that by the time of another Greek geographer, Strabo (about 60 years before Pausanias), there were three large tribes inhabiting Galatia (the place where the Celts eventually settled in Anatolia). These were the Tolistobogii (in Latin; Tolistoagioi in Greek), the Tectosages, and the Trocmi - read Strabo's description of these, and of their territory Galatia, here. Trying to identify individual groups of Galatians beyond that is a matter of detailed scholarly discussion not really necessary here, and even the process of how these three large groups formed is really pretty unknown, and the details of how they governed themselves are up for debate.
At the very least, we can learn from etymologizing the names of some of the Galatian tribes. Let's start with the Tectosages: one possible meaning for the root tecto- is related to Latin tectus, -a, -um, a word meaning 'covered' but often referring to a house. The component -sag- can be much more clearly determined to mean 'one who searches' - so therefore the 'Tectosages' are possible 'the ones seeking homes'. Not exactly a ground-breaking proposition, given the nature of their migration, but the fact that they would have this as an ethnonym (i.e. a tribal name) within their own language suggests that seeking a new home may have been of particular importance to that group. Let's compare this with the Tolistobogii (Tolistoagioi); the latter component of their ethnonym, -bog- means 'beaters, bashers' or 'fighters'. The meaning of tolisto- is unknown, but may have a connection to an old Irish word tol- meaning 'will' or 'desire' - so perhaps we have a group named 'those who like to fight', or something along those lines. In this case, their migration can perhaps be ascribed more to a desire to raid and plunder rather than make a new permanent home elsewhere.
Source for these etymologies: Xavier Delamarre, 2003. Dictionnaire de la langue gauloise. Only available in French, unfortunately, but such is the nature of studies on Celts (e.g. much scholarship on the Celts and Galatians in particular is available only in German).
For the above-mentioned seminar on Greco-Roman Anatolia, I explored the extent of Celtic settlement in Anatolia outside of the traditional borders of Galatia. Here is the paper I produced. My thesis is that there is evidence of Celtic settlement names outside of these traditional borders, suggesting a wider Celtic presence than one might believe; once again this speaks to the varied and independent nature of Celtic tribes, who appear to have settled a vast expanse of territory from Bithynia all the way to the mountains of Armenia, almost certainly over the course of a couple of centuries. Our knowledge of their settlement comes primarily from the literary record left to us once they had formed themselves into cohesive political entities (though even this process is obscure and a matter of debate) by the turn of the millennium, but I believe that these names reflect a much wider Celtic influence than is normally accepted.
This fierce people, after wandering and warring over almost the entire world, have taken up their abode amongst the gentlest and most peaceable race of men. Their tall stature, their long red hair, their huge shields, their extraordinarily long swords; still more, their songs as they enter into battle, their war-whoops and dances, and the horrible clash of arms as they shake their shields in the way their fathers did before them - all these things are intended to terrify and appal. But let those fear them to whom they are strange and startling, such as the Greeks and Phrygians and Carians.
Simply put, they were ferocious in battle, probably bigger than most of the Greeks or indigenous Anatolians they might face in battle, and had an incredibly fearsome reputation, meaning most would prefer to simply pay them to stay away rather than suffer the rape and pillage which the Celts would bring with them. Indeed, Attalus I of Pergamon seems to have based his claim to kingship on the fact that he refused to pay the Galatians tribute and defeated them in battle. (A detailed study of this evidence can be found in this book).
Like I said, this is a topic very close to my heart and I'd be happy to answer any further questions on the subject! Here's some further reading - my apologies that some of it will not be in English...
Darbyshire, Gareth, Stephen Mitchell, and Levent Vardar. 2000. The Galatian Settlement in Asia Minor. Anatolian Studies 50, 75-97.
Mitchell, Stephen. 1993. Anatolia. Land, Men, and Gods in Asia Minor. vol. 1: The Celts in Anatolia and the Impact of Roman Rule; vol. 2: The Rise of the Church. Oxford: Clarendon Press. (This is a seminal work on ancient Anatolia. I refer to it in my own research on a constant basis.)
Mitchell, Stephen. 2000. Ethnicity, Acculturation and Empire in Roman and Late Roman Asia Minor. In: Mitchell, Stephen; Greatrex, Geoffrey (eds.), Ethnicity and Culture in Late Antiquity, 117-150. Swansea: Duckworth/The Classical Press of Wales.
Stähelin, Felix. 1973. Geschichte der kleinasiatischen Galater. 2nd edition 1907. Repr. Osnabrück: Zeller Verlag.
Strobel, Karl. 1996. Die Galater. Vol. 1 : Geschichte und Eigenart der keltischen Staatenbildung auf dem Boden des hellenistischen Kleinasien. Berlin: Akademie Verlag.
If you are interested in the Germanic tribes during the early Roman empire, a good source would be Tacitus' Germania
Procopius, who accompanied Belisarius on his campaigns, also wrote about the Slavs during their migrations.
In early antiquity, there were many hundreds of thousands of Celts of in Europe, a large portion as far east as Hungary. The Celtic tribes, gradually expanded south, pushing other groups out of the way. They were strong fighters who could hold their own against other tribes, and moved south after displacing previous inhabitants. They could not expand very far to the east, as the warlike Dacians countered them.
In the 4rd century, one of the groups they came in contact with were the Illyrians, who lived between what is now Albania to the south, Croatia to the north, and Serbia to the east. The Illyrians had fought the rising power of Macedon under Philip I, greatly weakening themselves in the process. Celtic tribes took advantage of this to establish themselves in Illyrian territory, with access to Greece.
Greece was wealthy, with plenty of potential loot. However the Celts dared not touch Greece under Alexander, as the Macedonians were very formidable at this time. However, upon the death of Alexander, they found an opportunity to invade.
The division of Alexander's empire left the Greeks in constant warfare with each other over the division of its lands. Preoccupied with themselves, the Greeks did not prepare for a Celtic invasion. In a large raiding expedition against Greece and the rest of the Balkans in 279 B.C. with 80,000 men, the Celts managed to defeat Macedonian armies several times before they themselves were defeated.
10,00 men from that large army was hired by a Bithynian king, Nicomedes I, to secure his throne from dynastic disputes and Seleucid invasion. The Celts moved to Asia Minor as mercenaries for the king, then stayed afterwards, raiding the Hellenized inhabitants of the area. Successive migrations to Asia Minor by other raiding parties strengthened the Celtic position.
The territory they occupied became known as Galatia, which is similar to the Latin Gallia. The Celts hired themselves out as mercenaries to states in the area and were considered formidable fighters. They were opposed most notably by Pergamum, who detested Celtic raids on their lands. However, they were supported by other states, like the Seleucid Empire, which viewed them as a balance to enemies like Pergamum.
tl;dr They migrated south because they were strong fighters and were invited to Anatolia as mercenaries.
*edit- Added reading recommendations on the Germanic migrations.
Something to ponder about your question (if not your answer) - if you want to know why the "Celts" migrated so far to go to Anatolia, where do you think they came from? There is actually some healthy debate about who the Celts were (are Celts people who display "Celtic" art? Speak a "Celtic" language? Are called "Celts"? These groups actually include different people!) - you might be interested to have a look at these two works by John Collis, The Celts and The European Iron Age.
Like I said, there is an interesting debate on the matter, and some hypotheses still appear less likely than others (I went to a conference a few years ago where a speaker argued a Celticisation from the West, originating in Spain). These two books are good works to start getting your head around the matter.
