The Last Roman Emperor
The Visigoths did not stay in Rome, but marched down to the tip of Italy, apparently trying to get to Africa. But Alaric died there, and a big storm frightened them. In the end the Visigoths marched up to southern France and settled there. The Burgundians had already taken over eastern France, and the Vandals and Sueves were in Spain (the Alans got wiped out).
In 429 AD the Vandals sailed across the Straits of Gibraltar and took over Africa, again without much of a battle. This left the Sueves alone in Spain, and the Visigoths gradually began taking over parts of Spain. The Picts and other groups invaded England, and the English wrote to the Romans for help, but the Romans said they were on their own.
For more on the fall of Rome:
Classical Rome, by John Clare (1993). For kids, the whole political history from beginning to end.
The Romans: From Village to Empire, by Mary Boatwright, Daniel Gargola, and Richard Talbert (2004). Okay, it's a little dry, but it is up to date and has all the facts you could want.
Corruption and the Decline of Rome, by Ramsay MacMullen (1988). A leading historian argues that the collapse was caused by economic conflict between the upper and lower classes of the Empire. I don't agree, but see for yourself.
The Fall of the Roman Empire, by Michael Grant (1976, reprinted 1997) 0684829568. Grant's an easy, clear writer, but he blames the collapse on too much government bureaucracy and taxation - again, I don't agree.
Fall of the Roman Empire, by Arthur Ferrill (1986). Blames the collapse on military losses - I find this much easier to believe, although Ferrill's book emphasizes Roman military mistakes rather than the strengths of the invaders.
Early Medieval Europe, 300-1000, by Roger Collins (2nd edition 1999). Sees a gradual transition, rather than a collapse, but I like the emphasis he places on attacks from outside the Empire.
The World of Late Antiquity AD 150-750, by Peter Brown (1971). Also advocates for a gradual transition, with an intellectually thriving early medieval world.