What happened to the Gothic Peoples (Vandals, Visigoths, etc.) and their kingdoms? Did the Kingdoms just fall apart? Are there still Gothic people in these areas?

There are two major Gothic kingdoms that emerged at the end of the fifth century: the Visigothic Kingdom of Toulouse and the Ostrogothic Kingdom of Italy. I have written previously on the foundation of the Visigothic kingdom. To be brief, the group moved into southern France in the opening decades of the fifth century and began to make inroads into the Iberian peninsula, often under imperial direction; by the mid-sixth century, the focus of the kingdom had shifted to Toledo following the loss of territory north of the Pyrenees. The Ostrogoths under Theoderic left the Balkans for northern Italy in the late 480s, and by 493 had conquered the territories that had Odovacer had formerly ruled in the name of the Eastern Roman Emperor.

It's important to remember that these groups were a diverse bunch, so broad labels like Ostrogoth and Visigoth should be treated cautiously, particularly when discussing early migrations. These groups were also the minority in their newly conquered territories-- they never represented more than a small percentage of the population. The Ostrogoths more or less confine themselves to northern Italy. Their impact on the gene pool would have been minimal, but to an extent, yes, their descendants are still in Italy in some form or another, but to the extent that this is useful information is up for debate. Furthermore, ethnic identity does not derive from genes, and even during the tenure of their respective kingdoms, identity was fluid as Roman elements were adopted and incorporated into the rule of the Gothic elite. The notion of a Gothic heritage would be preserved by medieval writers in northern Spain when framing conflicts between the Christian kingdoms and Muslim Al-Andalus, but generally speaking, Gothic identity simply changed, merged, and disappeared following the conquest of the Gothic kingdoms and the social and cultural transformation it entailed.

The Ostrogothic Kingdom ended first. Following the long rule of Theoderic, internal conflict and intrigue, typical during the minority of a young king (Athalric was still a boy when he succeeded) weakened regnal authority. The intervention of the Eastern Romans and the subsequent Gothic War crippled Gothic rule and led to the collapse of the kingdom, as well as the re-integration of Italy into the empire. This was not to last however, as the Lombard Invasions beginning in the late 560s would see the old Ostrogothic power base in the north incorporated into a new 'barbarian' kingdom. I have covered the demise of the Visigothic Kingdom of Spain before here. As I mentioned above, some semblance of a Gothic heritage was preserved in the north, but to the extent that this represents real continuity, or an invented origin to legitimize conquest of former Visigothic lands is up for debate. It's certainly possible that some of the Visigothic nobility fled north, but some stayed, as evidenced by various treaties made between the conquerors and local magnates who retained their territories in return for acceptance of caliphal rule.

The Vandals were not Goths, though they are certainly part of the broader migrations of east Germanic peoples. They established a kingdom in North Africa in the early fifth century, and like the Ostrogoths, also fell to Justinian's reconquest of the old Roman territories throughout the Mediterranean. I do not know enough, however, to comment on what happened to the Vandal elite following the conquest; perhaps someone else could comment on that point.