What natural dyes were brought over from America in the years after the Europeans established routes and expeditions to the New World? Did the fashion become more colourful in the 16th century and were the dyes sought after in the same way as spices?

Dyes were brought over and sought after, but we don't see an explosion of color in popular fashion in response as 1) the New World dyes had to be imported at great expense and 2) most of the colors could be achieved to some extent with other dyes. The impact of these dyes was more in the money they brought to colonizers, the conflicts over and exploitation of the colonies, and the push/pull in European countries without the colonies between using new (imported, expensive) dyes vs. traditional (and often domestically-made) known quantities.

Probably the best-known New World dye is the "perfect red": cochineal, which was the name of both the dye and the insect that is ground up to produce it. The Old World had kermes and lac (similar insects that produced a not-as-great red) and madder (a plant that produced ruddy colors even farther from true red) and when cochineal was brought back it was a huge deal. Cochineal required a smaller amount of dyestuff for a pot, and the insect was much more plentiful than kermes or lac.

Brazilwood is a tree that likewise had an Old World relative and was found to be much more plentiful in South America. (Brazil is named for it, not the other way 'round.) The dye was extracted from the fermented heartwood, and could produce vibrant reds, pinks, and purples depending on the mordant used to set the dye in the fiber. Indigo has a similar story - it grew in Asia as well, but was plentiful in New World colonies, and Spain and Portugal were able to make a killing by selling it domestically rather than having to import it from foreign merchants.

Logwood was the other "big" New World dye. Also made from heartwood, it could produce yellows, reds, blues, purples, and black depending on the mordant used on the fabric. The black was especially important as it was otherwise a difficult color to achieve - other means produced colors that weren't quite a true black or resulted in the fabric decaying more quickly.