Are new connections continuously forged in the adult human brain, or do we just start all wired together and then prune until we have something we like?
Development of patterns of synaptic connection contains elements of both outgrowth and pruning. For example, this paper from zebrafish development illustrates both those mechanisms, which appear to be tightly linked:
Meyer MP, Smith SJ. 2006. Evidence From in Vivo Imaging That Synaptogenesis Guides the Growth and Branching of Axonal Arbors by Two Distinct Mechanisms. Journal of Neuroscience 26:3604–3614.
In more advanced animals (e.g. cats), there is also evidence for both outgrowth and pruning:
Callaway EM, Katz LC. 1990. Emergence and Refinement of Clustered Horizontal Connections in Cat Striate Cortex. Journal of Neuroscience 10:1134–1153.
There is also evidence for generation of new neurons in adult humans, which implies that new synaptic connections must be forged to integrate newborn neurons into existing networks:
Ericsson et al. 1998. Neurogenesis in the adult human hippocampus. Nature Medicine 4:1313-1317.
The phrase "fire together, wire together" comes from an explanation of Hebbian Learning and refers to the adaptation of synapses as a response to the firing of already connected neurons. This is one of several Synaptic Plasticity mechanisms. Two others that exist are Long Term Potentiation (strengthening and creation) and Long Term Depression (weakening and destruction).
All of these mechanisms are still being researched heavily to understand their detail better. However, I hope this gives you a good starting point.