In type IIA string theory, there are D0, D2, D4, D6, D8 branes, and in type IIB there are D1, D3, D5, D7, D9 branes. The D9 is a bit special because it is spacetime filling (p+1 = 10 for p=9).
In both theories there are NS5 branes and NS1 "branes", more commonly known as fundamental strings (i.e. the string in string theory). The brane content can be figured out in different ways, a way that comes from the low-energy effective supergravity theory was explained here Why M-theory has only M2 and M5 branes? (the explanation was given in the context of M-theory but it holds for the 10-dimensional case as well).
The point of the above was to clarify that NS branes only exist for p=1, 5. As you point out, the NS5 and D5 are S-dual. It's also true that the NS1 and D1 are S-dual. Now if you want to find a connection with an NS-p brane and a D-p' with $p \neq p'$, you'll need to use T-duality in addition to S-duality. T-duality changes a D-p brane in one theory (either IIA or IIB) to a D-(p-1) brane in the opposite theory. So, for example, we could take the following duality chain to go from a D6 brane in IIA to an NS5 brane in IIB:
$D6 \rightarrow_T D5 \rightarrow_S NS5$
Here the subscript denotes the duality transformation applied.
EDIT: As to the connection with $p$-branes in supergravity, both the NS- and Dp-branes have "p-brane" supergravity dual solutions. By the way, the way you tell if you should think of the object as a solitonic brane in string theory, or a warped p-brane metric in supergravity, is whether or not the backreaction on the background space is significant. This is controlled by the parameter $g_s N$. When this is small, the solitonic brane description is weakly coupled, and when it is large, the supergravity description is valid.
If you're interested in how the two descriptions relate, there is of course the AdS/CFT correspondence which relies heavily on this connection. Also, the famous microstate counting of Strominger and Vafa relied on this as well.