The history of the orchestra is a pretty big part of music history that spans a couple of centuries, but the modern instrumentation of the orchestra was more or less solidified by the late Romantic period. I'm going to approach this by talking about the major composers of those Classical (roughly middle of 1700s to early 1800s) and Romantic (late 1700s to early 1900s) periods. Many of the works I am going to reference can be found on the IMSLP sometimes even with copies of the original scores when available.

Before these periods, brass and woodwinds weren't as regularly used in orchestral music as the strings. Natural trumpets and natural horns (those instruments without valves) were regularly used through the Baroque and Classical periods. J. S. Bach wrote for the natural trumpet in the Brandenburg Concerto No. 2 and for the trombones in a few sacred works.

The style called the Symphony later became a staple of the Classical and Romantic periods. Many of Haydn's symphonies are scored for strings, two oboes, bassoon, and horns. By the end of his career he usually included two trumpets and timpani as well as flutes and clarinets. Mozart was similar, with early symphonies sticking to strings and double reeds while later works added horns, trumpets, timpani, and clarinets.

Beethoven slowly expanded the brass. His early symphonies are scored roughly the same as late Haydn and Mozart symphonies. Symphony No. 5 is notable for the adding trombones, which had beforehand been largely absent from secular music. Symphony No. 9 included a piccolo, more percussion, a large brass section and a choir.

After Beethoven, the development of the valve for brass instruments allowed them to be used more extensively that the limited natural trumpets and horns. In 1844, composer Hector Berlioz published his Treatise on Instrumentation. An English translation of a later edition can be found here. Although the instruments he writes about weren't always used in every orchestra, the treatise gives us an idea about what instruments were available and what their roles were.

The later Romantic composers largely scored similarly to Beethoven. Brahms' four symphonies are all scored roughly similar to Beethoven's. Mahler, writing in a post-valve world, would add more. His scores often include more woodwinds and percussion, sometimes including harp. We could jumpinto works by a great number of romantic and modern composers, but by this time the orchestra at is core is mostly what we are now familiar with.

Worth noting is that some instruments described by Berlioz rarely are seen in the orchestra. The Saxophones and Saxhorns, for example, are usually absent, although the Saxophones were adopted by jazz and the Saxhorns became popular in civil war era regimental bands.

See also, Cecil Forsyth's Orchestration, here.