You originally had asked a very broad question, so I'll try to demonstrate why that is such a hard question to answer. I've done two fairly large differential analysis studies (and a few smaller ones) covering very different areas of research, and the approaches that other researchers used subsequent to my differential expression calculations were unsurprisingly also very different.
In the first study, I did a de-novo transcriptome assembly of Schmidtea mediterranea (Smed) from Illumina RNASeq reads, then carried out a DE analysis (DESeq) on a β-catenin knockout mutant population compared to wildtype. The other researchers had some very impressive tools available at their disposal, allowing them to subsequently visualise the positional expression of differentially-expressed genes in the knockout and wildtype populations after amputation. They found a few candidate genes that had graded expression along the anterior-posterior axis in the wildtype population, and this expression disappeared in the knockout mutants. One of these genes was a teashirt (tsh) family member, and a good candidate for further discovery based on the researchers' existing knowledge of the gene pathways associated with cellular regeneration. To cut a very long story short, they experimentally validated the teashirt link through further RNAi studies, generating a double-headed Smed as a result. This was followed up in Zebrafish (Danio rerio), because there happened to be another building about 100m away that specialised in Zebrafish development.
The second study was/is a bit more similar to your current situation. We looked at sorted dendritic cell expression in mice after the application of two Th2-inducing treatment conditions (injected Nippostrongylus brasiliensis; applied DBP+FITC) and one control condition (injected PBS). I did a differential expression analysis with DESeq2, and developed a Shiny App for result exploration after I got tired of getting asked to do a whole bunch of mechanical follow-up tasks (mostly generating graphs, lists, and heatmaps for different gene subsets). We ended up with thousands of differentially expressed genes, so it wasn't feasible to look individually at every gene for research evidence and pathways -- one scientist spent a couple of weeks looking at one interesting-looking candidate gene. The differentially-expressed genes were fed into the oh-so-expensive IPA (which gives results, but not necessarily good ones) and DAVID, among a few others. These helped a little bit, and led to what was the central idea of our paper (i.e. that there were at least two different Th2 responses), but we still had the problem of too many "significant" pathways, and lots of differentially-expressed genes that were members of a whole bunch of other pathways. We tried WGCNA; we did a bit of GSEA; and we compared Limma and DESeq2, but still couldn't find anything concrete or obvious to demonstrate what was triggering the Th2 response. There were experimental studies as well: blocking and mutant studies to investigate the two types of Th2 response (and experimentally confirm the differential expression). Our experiments and exploration are still ongoing, but we felt it was necessary to get one paper out at the intermediate stage, so just ended up taking the top 25 differentially-expressed genes in each experiment (by fold change), and finding some way of grouping some of them together (via expert knowledge and literature searches). It kind-of-sort-of worked, but we're left with a bit of frustration in that we (and the tools we used) were basically stumped by having too many biologically-relevant results.