What did indigenous North Americans trade amongst themselves before Europeans arrived?

An area which may require a closer look is the southwestern cultures. Trade routes were actually quite well established in these regions, with many items being traded. An article Indigenous Trade: The Southwest , lists many of these trade materials(emphasis highlighting trade materials mine):

Anasazi. Around the end of the first millennium a.d., Anasazi Indians living in the Southwest had become fully integrated into the pansouthwest trade network. They supplied highly valued turquoise and, to a lesser extent, obsidian to tribes located along the Gulf of California in exchange for luxury goods such as bracelets and pendants fashioned from Pacific shells. They also traded turquoise with Mesoamerican civilizations such as the Toltec Empire for high-prestige items such as macaw feathers, ornaments, and pottery. This intercourse had important consequences because it helped spread Mesoamerican pottery styles, religious customs, crops, and agricultural techniques to North America.

The next section discusses a later time, and repeats some of the information included in Lars answer concerning the trade between the Pueblo and Plains cultures, but I'll again include it to highlight the materials being traded.

New Avenues. After the pansouthwest commercial system collapsed between 1200 and 1400, the pueblo-dwelling Indians of the Rio Grande valley began to trade with semisedentary plains tribes such as the Apache. Pueblo tribes such as the Tewas exchanged surplus corn, cotton textiles, ceramics, and turquoise for the Plains Indians’ tallow, salt, buffalo meat, and hides. This new commercial intercourse was based, in part, on the same system of reciprocal gift giving that governed trade among the Indians of eastern North America. Commerce between Pueblo and Plains tribes was substantially more complex than reciprocity-based trade, however, because it involved the complementary exchange of surplus goods. It thus allowed the Plains tribes and, to a greater extent, the Pueblo Indians to shift from a simple, subsistence-based economic system to a more complicated one based on specialized production.

By some accounts, oils and furs were traded in the Pacific Northwest.

Early accounts stress the enormous importance of oils in trade, feasting, and food. The Makah used to compete to see who could drink the most whale oil at feasts (Colson 1953). People were desperate for oils. Suffice it to say that “ooligan” is derived from a Tsimshian word meaning “savior,” now used for Jesus. Watertight boxes of oil from the ooligan (oolichan, eulachon), a smelt that is mostly fat by dry weight, were traded all up and down the coast. The Haida sailed their great canoes over tens of miles of some of the most dangerous waters in the world, and traded their most valuable possessions, to get these boxes of oil. Northwest Coast: Traditional Indigenous Relationships with Plants and Animals

Non-residents of the Nass (i.e. Non-Nishga cultures) journeyed from the interior early in the year, while the snow was still deep in order to reach the Nass River for fishing time (mid March). They traveled hundreds of miles with their belongings on sleighs drawn by dogs or themselves. The non-Tsimshian among them also brought furs (usually marmot and rabbit skins, but also martin, mink, and bear skins), to pay the resident cultures of the river for fishing rights and to pay them for using their nets and shelter in their fishing lodges.

In regions of Coastal British Columbia where there were no eulachon, the people obtained them through trade, usually in the form of eulachon oil [62]. Eulachon oil was so highly prized by many cultures that the Northwest Coast cultures traded it long distances eastward to cultures in the interior along so-called “grease trails”. There are several ancient trade routes to the coast called “grease trails” and the great explorer Alexander Mackenzie made his famous journey to the Pacific ocean following one such “grease trail” [5]. The First Nations of the Northwest Coast traveled the grease trail into the interior to trade the oil with the Athapaskan-speaking tribes and they traded the oil by canoes to the south and north 1. The Gitksan Tsimshian, who had a winter village on a grease trail to the Nass, traded soapberries, dried fish, meat and tanned hides to the Niska of the Nass for eulachon [30, 58].

Eulachon

Flint points, obsidian, buffalo hides, salt, pearls, shells and (as mentioned by T.E.D.) copper were among item traded by Native Americans before Europeans arrived. That said, trade in North America prior to European contact varied greatly in its extent and volume, depending on area and epoch.

From Trade Among Tribes: Commerce on the Plains before Europeans Arrived by Samuel Western (Wyoming State Historical Society), our two main sources of this trade come from archaeology and from accounts of the first European traders to reach various regions.

From archaeology,

Indians of the southern and northern Plains traded with each other for thousands of years. Flint points 13,000 years old, chiseled from the Texas quarries, have been found in eastern New Mexico. Quarried stone from the Obsidian Cliffs near Mammoth Hot Springs, Wyo. in Yellowstone Park, traveled to the the Ohio River Valley around 100-350 CE.

.....

Archeological artifacts do suggest...that native-to-native trade expanded over time. Emory Dean Keoke and Kay Marie Porterfield, authors of the Encyclopedia of American Indian Contributions to the World, say that the Hohokam tribe, centered in present day Arizona, traded seashells, which they had acquired from the Mojave tribe, for buffalo hides from various southern Plains tribes. “By between 500 and 200 B.C., North American Indians had established a vital network of trade.”

Also,

A research team including Matthew Sanger, assistant professor of anthropology at Binghamton University, State University at New York, has found a copper band that indicates ancient Native Americans engaged in extensive trade networks spanning far greater distances than what has been previously thought.

The researchers found that the band was made from copper that

originated in the Great Lakes region, more than 1,500 km away. Copper sources each have their own unique chemical makeup, including very small amounts of trace elements. As such, archaeologists can match manufactured objects to their sources by comparing their chemical signatures,...

Copper was also traded in the Southeast (among other regions), along with salt, pearls and probably ceramics. Many kinds of stones / minerals (sandstone, soapstone, chert, galena etc.) were also traded, both before and after their transformation into status items and / or weapons.

On food (other than corn / maize), the sources are less definite. On the one hand,

Because North America lacked an animal that could be domesticated for draft purposes and was wanting for any other efficient means of regional and continental transportation as well as an organized market system, the people of this continent did not trade food products to any great extent.

Source: 'Chapter 1' in Jonathan E. Ericson & Timothy G. Baugh (eds), 'Prehistoric Exchange Systems in North America'

On the other hand, it is impossible to rule out local exchange of foodstuff. Michael B. Stewart, in Late Archaic through Late Woodland Exchange in the Middle Atlantic Region (in Ericson & Baugh) states:

Foodstuffs and artifacts fashioned from organic materials are poorly preserved in the archaeological record of the region, their absence serving to remind us that we are seeing only a part of the whole picture of Native American material culture.

Nonetheless, Susan C. Vehnik and Timothy G. Baugh in Prehistoric Plains Trade (in Ericson & Baugh) cite Henning (1983a) as suggesting that there was some trade in bison hides and dried meat in the period 800 to 1200 AD, while trade in 'bison products' is mentioned in relation to the Southwestern Pueblo in the period up to 1650. Given that Native Americans could and did hunt bison before they had horses, the trade in items derived from these hunts is at least plausible.

Accounts of early European traders are also a valuable source of information as

artists who visited the upper Missouri and Rocky Mountains in the 1830s noticed tribes hanging onto traditions or only selectively using European goods.

The addition of European goods did not suddenly change Native American trading habits, although it should also be pointed out that tribes sometimes acquired European goods from other tribes (i.e. without meeting Europeans themselves). Tribes

tapped Wyoming’s abundant natural resources for desired trade goods: quartzite or obsidian for knives, scrapers and arrowheads; buffalo for robes, dried meat, pemmican and hides; soapstone for bowls; elk or deer for tanned hides; and horn, particularly from the bighorn sheep, for making bows, which were highly desired.

"Trade links among northern Plains tribes about 1775, before the arrival of Europeans. Courtesy W. Raymond Wood." Source: WyoHistory.org

Western notes that:

The Shoshone, it seems, traded with everyone, including northwest and southwest tribes. Other Rocky Mountain and central Plains tribes also took goods to the Missouri River valley to trade for corn, pumpkin, squash and native-grown tobacco (Nicotiana quadrivalvis, Pursh)...

Over time, European goods were traded alongside / in exchange for local products:

Corn also appealed to former woodland tribes. “For the Sioux, corn was more important than blood,” says James P. Ronda, professor of Western American History at the University of Tulsa. In August, “as in every other late summer and early fall, Sioux bands flocked to the Arikara towns, bringing meat, fat, and hides from the plains and European-manufactured goods from the Dakota Rendezvous.”