I'm collecting the comments together into an answer:
There seems to be no question that Pete L. Clark's Notes on Convergence are a very thorough treatment of nets, filters and convergence. There are also a few exercises on summation (which always is absolute summation).
In my experience there's no more need for basics than what's contained in chapter 1 of Pedersen's Analysis Now.
Two classic references are J.L. Kelley's book General Topology and R. Engelking's General Topology.
Michael Greinecker and Martin Sleziak both recommend Eric Schechter's Handbook of Analysis and its Foundations for a detailed discussion of the various notions of subnets that can be found in the literature. Parts of this is contained in these notes and these by Martin Sleziak.
Furthermore Aarnes and Adenæs, On Nets and Filters, Math. Scand 31 (1972), 285–292; was recommended.
Basic results on nets are given in Chapter 2 of Megginson's book An introduction to Banach Space Theory, Springer GTM 183. This book also covers nets in topological groups (starting from p.154) and topological vectors spaces (starting from p.167).
These aspects of nets might be interesting to you if you plan to use nets in functional analysis. From the above mentioned books both Pedersen and Schechter are oriented towards analysis, too.