This is a really, really big question. As in, thousands of books have been written about South American societies during this time period, and there are thousands more to write. So I'm just going to give some highlights and tidbits - feel free to ask about specifics in a reply! I'm going to generalize the period you're asking about to the timeframe from around 100 B.C. to 500 A.D. I'll also divide this up regionally. And I probably wrote too much, but it feels like too little.
Ridiculously short overview of what I wrote: Lots of cities and the beginnings of big empires in the central Andes, towns and metallurgy in northern Andes, mostly hunter-gatherer bands in the far south and southeast, mystery in the Amazon.
The Andes and West Coast
This is the region that we know most about during the given time period. By 100 B.C., the central Andes and coast (centered on what is now Peru) had already experienced millennia of cities, states, and urban polities. Three of the most famous historical South American civilizations were thriving or getting started right during your time period: the Moche, Nazca, and Wari. There is debate about whether the Moche were a state, a couple states, or a culturally related association of city-states, but whatever the truth, this society thrived on the northern coast of modern-day Peru from 100 to 700 A.D. Moche leaders created cities dependent upon irrigated agriculture with massive monuments, like the Huaca del Sol. They worked metal - especially gold, silver, and copper - into massive crowns and other ornaments. And they are perhaps most famous for their incredibly varied and skillful ceramics, which ranged from highly naturalistic animals to mythical figures sculpted and painted in remarkable detail.
Along the southern coast of modern-day Peru, the Nazca civilization flourished from 100 B.C. to 800 A.D. Although different from the Moche, the Nazca likewise lived on an extremely arid coast and depended on intensive irrigation channels to maintain cities. Like most Andean civilizations, pottery and textiles were important to the Nazca, but Nazca textiles are recognized as some of the most complex ones made. And, most famously, the Nazca created the famous Nazca Lines: giant geoglyphs of birds, trees, people, and lines in the desert that were probably walked along during rituals.
By 500 A.D., when the Roman Empire was collapsing, one of the Andes' largest Empires was just beginning to grow. Although very little is know about their early history, the Wari Empire eventually controlled a large part of the central Andes, and created a centralized state that foreshadowed the eventual Inca Empire in many ways. Like the Inca, the Wari used the quipu recording system based on knots, maintained extensive roads, and built forts and administrative centers throughout its lands. One of these administrative centers, named Pikillaqta, was eventually built in the valley that would come to host the Inca capital centuries after the Wari's collapse around 1000 A.D.
I'll end my Andes/western coast highlights here, with one example of the many smaller, less well known societies which were scattered throughout the area and aren't known to have created states or more unified societies. One of the Lima culture's most extensive sites is Pachakamaq, a massive collection of pyramids and monuments located just outside of today's Lima. This city served as a pilgrimage center for the entire region for more than a thousand years, and is an excellent reminder that urban societies would have been common throughout Peru during this time period, drawing people and resources from faraway places, even if they were not parts of large, definite empires. There were many places like this, and I've mentioned only the barest handful.
The North
I'm kind of cheating here - I'm going to talk about another Andean society, but I'll slightly excuse myself by saying that the far northern Andes are generally considered differently from the rest of the massive mountain range. The societies of most of what would eventually become Colombia and Venezuela were different from their counterparts in the more southern Andes, and many were more directly related to or in contact with peoples from the Caribbean and Central America. One of the more impressive societies of this region reached its height around 300 A.D. The Quimbaya people lived in modern-day Colombia, and are well known for their metalworking. As one of the earlier cultures to make use of the "tumbaga" gold and copper alloy which would become popular throughout the region, the Quimbaya developed a careful metallurgy tradition, making use techniques like lost-wax casting and depletion guilding. Other societies that existed in the area at the time, which were in many ways comparable to the Quimbaya, included the Tairona and Calima cultures.
The South
Southern South America - now most of Argentina and Chile - isn't known to have developed more urban and populated societies as much as other parts of the continent. This might have to do with the area's rough climate. Most knowledge about this area concerns hunter-gatherer societies, especially during the time period you're asking about. However, in the early centuries A.D. the El Molle culture would have been growing in what is now northwestern Argentina and northern Chile, and it would eventually give way to cultures that we can confidently say were sedentary agriculturalists.
The Amazon
The Amazon Rainforest might have the most mystifying history of South America - and, given how little we know about the continent's deep past, that's saying a lot. Much of the rainforest's current depopulated and wild state is a result of post-European contact societal collapse and the jungle's ability to grow back. But modern research (and deforestation) is increasingly showing that the Amazon was home to many, including more sedentary societies. Things like concentrations of useful plants, orchards, the near-magical terra preta soil that communities created, and massive geoglyphs are only beginning to show us how anthropogenically constructed some parts of the Amazon were. In fact, the earliest known invention of pottery in the Americas comes from the Amazon, made around 6000 B.C.
Around 500 A.D., the Marajoara culture would have been just beginning at the mouth of the Amazon river. Known for their striking and often huge pottery, this stratified society is one of the few certain examples of historical Amazonian societies that we have
The Southeast
This short section is a miniscule overview of what is now Brazil south of the Amazon, and Uruguay. We know little about this area, and I hate to generalize, but I can't say much more than we know that much of this land was home to hunter-gatherer bands and societies. Along the coast of Uruguay and Southern Brazil, some fishing communities seem to have created massive shell mounds that probably served as building foundations, but very little is known about these cultures.