How can I analyze a set of DEGs (differentially expressed genes) to obtain information from them?

There are a number of questions that you may answer using your set of DEGs. These include:

1) What are the biological process, cellular locations and molecular functions that are particularly over- or under-represented in your set of genes.

2) What are the pathways that are significantly impacted in your condition?

3) What are the putative mechanisms that would explain all observed gene expression changes in your system?

4) What are the miRNAs that could be important in this condition based on the gene expression changes observed?

Here are some details about each of these:

1) What are the biological process, cellular locations and molecular functions that are particularly over- or under-represented in your set of genes. This is generally referred to as GO analysis. There are many tools that allow you to do this but there are also many, many mistakes related to the way the p-values are calculated and corrected for multiple comparisons, the association of genes to multiple GO terms and the implicit redundancy, the choice of the background set of genes, etc.. You may find this Nature Reviews Genetics paper useful: "Use and misuse of the gene ontology annotations" http://www.nature.com/nrg/journal/v9/n7/full/nrg2363.html

2) What are the signaling pathways significantly impacted in your phenotype (or significantly different between the two phenotypes compared)? There are several tools that do this kind of analysis. They can be divided into several categories:

2a) simple look up into pathway databases (eg. Reactome, KEGG, etc.). Here your set of DEG will be mapped on pathways. No real analysis is performed but you see what DEGs are on each pathway. The limitations are obvious: no p-values are calculated, no idea about which pathways are affected beyond random chance, etc.

2b) enrichment analysis of pathways. Here, pathways are considered as simple sets of genes and an enrichment p-value is calculated for each (e.g. DAVID, GSEA, Ingenuity, etc.). The limitations include the fact that the p-values are calculated based on the assumption that all variables (genes) are independent while the pathways are there precisely to tell you how these genes influence each other. Another limitation is that the pathways are treated as simple bags of genes, disregarding all the phenomena and interactions between genes that they describe. This analysis approach only looks at the number of DE genes and makes no difference between a situation in which a pathway has 3 entry points and all 3 are severely down-regulated thus effectively shutting down the entire pathway and a situation in which 3 other random genes are down regulated on the same pathway. See this Genome Research paper for instance about the limitations of the enrichment approach: http://vortex.cs.wayne.edu/papers/Genome_Research_reprint1537.pdf

2c) functional analysis of pathways (iPathway-Guide, ROnto-Tools). Here, the topology of pathway is fully taken into consideration and the impact on the pathway is calculated by propagating the signals through the pathway topology, as well as by considering the number of DEGs. See chapter 28 in here for details: http://www.amazon.com/Statistics-Microarrays-Bioconductor-Mathematical-Computational/dp/1439809755/ref=sr_1_1?ie=UTF8&qid=1422476470&sr=8-1&keywords=sorin++draghici

Techniques that address the last two questions (putative mechanisms and miRNAs) are not well documented in literature. iPathway-Guide (http://www.advaitabio.com/ipathwayguide.html#page=page-1) does offer both. Some of the sample datasets available for it are coming from phenotypes created by knocking down a specific gene. The tool is able to correctly identify the mechanism as a chain of gene expression changes that starts with the knock-out gene. In another sample data set, the phenotype is creating by treating with the mimic of a specific miRNA. Even in this case, the tool is able to correctly identify both the mechanism as well as the specific miRNA. so it's pretty interesting. The tool is commercial but one can do an unlimited number of analyses for free. The results are fully accessible for a few days. There is a charge only for storing the results over a longer period of time.

In case you prefer research software or a DIY implementation here is a review paper discussing over 20 topological pathway analysis approaches: http://journal.frontiersin.org/Article/10.3389/fphys.2013.00278/pdf