Oratorios are similar to operas, but are unstaged, and are based on religious topics. Some common examples include: Handel's Messiah, Saul, Israel in Egypt, and Esther; Bach's Christmas Oratorio (and, arguably, his two Passions, which are essentially similar in form), Haydn's Creation, and Mendelssohn's Elijah and Christus. The best thing you can do is "dig into" these, as you say, and read through a score while listening. I'd imagine the score for each of these can be found on IMSLP, and recordings can no doubt be found on youtube.
Oratorios consist of a large number of movements (sometimes arranged into parts -- Messiah is in three parts, for example). These movements can take a number of forms, similar to those in an opera. The most common are:
Recitative: These are essentially sung narrations, and tend to be fairly short, and freeform (no repeating). There is usually only one singer (at least at any given time -- sometimes singers with different roles might alternate in a sort of conversation), who sings an almost-non-melodic expository line. The accompaniment is likewise sparse, typically consisting (at least in the Baroque) of merely a Basso Continuo line, such as cello and a small organ or harpsichord, playing simple sustained chords. When there is an orchestral accompaniment (such a movement is sometimes called an "Accompagnato" instead of a "Recitative") it is usually light, and uses a similar "simple chords" style.
Aria: These are your songs. They are typically written for a single voice, and, unlike recitatives, are very melodic and virtuosic. If the recitative is describes the actions and dialog, the aria pauses the action and expands on an individual's thoughts. Often you will find some solo instruments accompanying the solo voice, and they may even toss melodic fragments back and forth between them. Virtuosic instrumental sections might get imitated by vocal melismas, and lyrical vocal melodies imitated by the instruments. At least in the Baroque, these are usually written in Da Capo Aria form, which is a version of ternary form (ABA). You have an opening section that conveys a single mood (whether joy, sorrow, regret, etc...), and which comes to a complete cadence, and is followed by a middle section with different text, typically in a different key, possibly with a lighter instrumentation, conveying a different but complimentary mood (sort of like a "bridge" in pop music). Then you go back to the beginning section and repeat it. In some instances, the closing section will be a shorter (or otherwise modified) version of the opening section.
Chorus: This is where you get to "go big" and include the full orchestra and choir. These often include at least some contrapuntal technique, and may even be written as a full-blown fugue (numerous examples in Bach and Handel).
These three elements will form a sort of backbone to the work, and even movements that don't conform exactly to one of them can often be described as being similar to one or more of them (for example: you might find an Aria in which the solo voice alternates against little bits of a chorus). In addition to these three common elements, you might have some additional types of movements:
Chorale: A favorite of Bach's (and a largely German/Lutheran tradition), this consists a simple hymn tune in a strictly four-part setting. Although the orchestra accompanies the choir, there's not usually any independent orchestral parts -- they merely double the choir. There's no technical virtuosity here; you could imagine a congregation singing along with these.
Arioso: A shorter, simpler version of an aria. Probably doesn't contain nearly as much virtuaosic material, and likely won't have the middle or closing sections of the Da Capo aria. OTOH, it is more melodic (and probably longer) than a typical Recitative.
Duets, Trios, Quartets: These tend to be rarer, and can be similar in structure to Arias, Ariosos, or choruses.
Instrumental movements: Often, the opening of an oratorio will be an Overture (but not always: Bach's Passions and Christmas Oratorio start with Choruses). There also may be other brief instrumental interludes, such as the short "Pastoral Symphony" movement in Messiah (which is not a true Symphony, in the classical sense) used to set the mood of shepherds in the field.
These are the basic types of movements you will see, but to see how they get used and combined, and what kinds of text and music gets used in each, you should really examine some actual Oratorios. Look for things like the sequence of movements, the structure of each movement, the texture, and the instrumentation. Also look to see how each of these serves the text. Between Youtube videos and IMSLP, there's a greater opportunity now than ever before for anyone to critically study these works, which is the best way to learn. There's really no excuse not to.