In my opinion, the Young Wizards series has a very loose continuity where you can pick up most of the books and understand without much background needing to be provided. That said, Deep Wizardry did have some significant developments for the series.
- Accepting failure - A significant part of the plot climax is that Kit and Nita have to accept that not only can they not always save everyone, but also that they cannot necessarily be able to choose that they will be sacrificed instead of others.
- Accepting death - This is a theme which happens repeatedly in the series, that while the Lone Power introduced entropy and death into the world, and they oppose him, it is now a natural part of the world, and attempting to prevent or hold off death can create additional entropy and suffering in the world.
- Maturing physically - This is a minor plot point in my opinion, but a definite evolution in the series. Kit and Nina start noting each other as members of the opposite sex.
- Maintaining the masquerade - This is also the first book in the series where the parents start realizing that their kids are up to something, even if they don't quite recognize what yet (and suspect that it has to do with the physical maturation above).
As noted by eshier, people and events in the second book do show up in later books, but my memory is that the references are self-contained enough that you will not be missing significant context.