Ignoring the aspect of the question that deals specifically with symphonic forms (which Mark Lutton has already answered), the answer to your question depends somewhat on what you consider an "orchestra".
In the Renaissance, musical parts were written out in full polyphony, but instruments weren't usually specified. The same piece could be sung by an ensemble, played by a viola consort, or a recorder consort, or any number of other combinations of instruments. Or one line could be sung, and the others played on a lute or keyboard. Church music was often accompanied by cornetts, sackbuts, and organs. But these typically act as doubling instruments, and do not have independent parts.
When we get to the Baroque, and look for genres that include voices and instruments, we immediately come to opera, the invention of which has been claimed to mark the beginning of the Baroque.
The first opera, Dafne, was written by Jacopo Peri, and first performed in 1598. Although it is now lost to us, it was apparently scored for harpsichord, lute, viol, archlute, and recorders.
The first surviving opera, Euridice, also by Peri, was first performed in 1600. According to the book Euridice: An opera in one act, five scenes, Peri lists four instrumentalists (playing harpsichord, lira da gamba, bass lute, and chitarrone [theorbo]) in the Preface, but does not imply they were the only ones. The book suggests that other instruments must have been used -- probably recorders, and perhaps violins -- but admits "we shall probably never know the exact composition of this first opera orchestra, nor precisely how the instruments were used."
Another opera named La Dafne was written by Marco da Gagliano and first performed privately in 1608. I don't know what the orchestral forces for this opera were, but wikipedia mentions that, in his introduction, "Gagliano recommends clearly separating the soloists from the chorus, [and] positioning the orchestra in front of the stage so the singers can see them properly," indicating that there was some sort of combination of chorus and orchestra.
These are all mentioned as precursors to what is probably the best answer to your question -- Claudio Monteverdi's opera L'Orfeo (youtube with score), which premiered in 1607, was published in 1609, and is the earliest opera that is still frequently performed today. In it, Monteverdi lists out very specifically some 40 instruments that he wants used in it's performance (though not all at once, so the number of instrumentalists can be smaller). These include various groups of strings, brass, winds, and continuo instruments.
The staves don't generally name specifically which instruments play which staff, but many of the movements have headings that indicate which instruments are to accompany the voices. For example: "This ballet is sung to the tune of five violas, three theorbos, two harpsichords, a double-harp, a bass viol, and a sopranino recorder." Or, "Charon sings to the sound of the regal" (a regal was a type of small reed organ). There are also specific instrumental obbligati, such as calling for a pair of violins or a pair of cornetts to play specific figures that are independent from the voices. In addition, there are several instrumental-only movements interspersed throughout (including, ironically, some that are labelled "Sinfonia", in accordance with the early usage of that term that Mark mentions).
Given the size and diversity of the forces involved, and the relative specificity with which they are employed (compared to other works that preceded it), I'd have to say that L'Orfeo is probably about the closest you'll get to a "first" work for an actual orchestra, as opposed to an ensemble. And since it also includes a choir, this piece is probably the answer to your question, or as close as you're likely to get (certainly within a decade).