At the time Haydn premiered his oratorio The Creation (in German, Die Schöpfung) in 1799, he had already achieved an iconic status, and was now considered the greatest living composer. Mozart had died nearly eight years earlier, and Beethoven had, in the past several years, just begun to establish his own reputation.
With a large-scale, multi-movement work for soloists, chorus, and orchestra, it's difficult to answer the specifics of your question without a detailed analysis of each movement, which is beyond our scope here. However, we can still provide some basic information. One way to understand The Creation is by learning about the general characteristics of what we call the Classical period in music (usually regarded as from about 1750 to 1825), and also to compare and contrast Haydn's oratorio with similar works of other composers.
An oratorio is a large vocal-orchestral work, usually, but not always, associated with a religious text. In the standard repertoire, the most performed and best-known piece in the genre is probably Handel's Messiah, which uses a text drawn directly from the King James version of the Bible. Haydn's The Creation uses the biblical text from the book of Genesis in alternation with additional text written by the diplomat and author Gottfried van Swieten, who had been a friend and patron of Mozart as well as Haydn. We can also note that Handel is a Baroque composer, prior to Haydn and the Classical period. In Handel, and much more so in his contemporary J.S. Bach, musical textures are more "complex" than in the later, more streamlined Classical style, though we'll see below that this generalization doesn't always hold true. One simply has to listen to the music to grasp these stylistic and period differences.
The best way to describe the music of The Creation is that it consists of an alternation of choruses with orchestra, arias and duets in which a soloist or soloists sing with orchestral accompaniment, and passages in which the orchestra plays alone. In between these set-piece "numbers" (as they were called at the time--a term that survived into the twentieth-century as a colloquial word for a popular song) are recitatives, which are a not fully melodic type of declamatory singing by a soloist, a kind of half-speech, half-song style in which the less poetic, purely narrative parts of the text are delivered.
Though by saying this we are oversimplifying: the texture of Haydn's music is mostly homophonic (meaning a melody with chordal, largely non-melodic accompaniment) with, as in the late-period music of Mozart, frequent interpolations of highly complex polyphonic passages (in which there are multiple melodies at the same time) reminiscent of Handel, Bach, and the Baroque period in general. Though Haydn's melodic and harmonic structure is relatively diatonic (based on the major or minor scale—the simplest way to explain this is that if the key is C, there will be relatively few sharps and flats) with not infrequent chromatic passages (in which the notes of the scale are altered by sharps and flats). Chromaticism—the added complexity of altered note—is used by composers for emotional effect and to express tension and uncertainty. In The Creation, this is most in evidence in the opening section, the depiction of chaos before God created the world. The music is highly unstable and the actual tonality or key is in doubt, until finally there is a clear C major chord at the words "And there was light." This kind of contrast was typical of the way composers in the Classical period depicted different meanings and emotions in a vocal text. Yet throughout The Creation, the dominant mood is the positive one of portraying the world, and then Adam and Eve, in a state of perfection before the Fall. This could be what prompted the German Romantic writer E.T.A. Hoffmann to comment that Haydn's music sounds as if it were written before Original Sin (though Hoffmann revered Haydn, as he did Mozart even more, and used the term "Romantic" to describe them).
In this last point we can grasp a major feature of not only The Creation but Haydn's work in general. Again, at the risk of oversimplifying, I would observe that in comparison with his younger contemporary Mozart, and with his pupil and successor Beethoven, Haydn's music does tend to sound generally cheerful and, perhaps even in the passage mentioned above in which "chaos" before the creation is portrayed, not focused on darker, more complex, or negative emotions. The Creation, especially, is a triumphant-sounding work that celebrates religion and tradition, perhaps ironically so given the turmoil and violence overtaking Europe at the time it was written, in the aftermath of the French Revolution and during the Napoleonic wars.