there is "wrong" or "right" answer on functional enrichment analysis, as there are numerous tools and methodologies, that test similar or different scenarios. So, each lab or group of researchers, following their own approaches for the biological translation of their bioinformatic findings. For example, if you have the normalized expression matrix of all of your genes (i.e. microarrays or RNA-Seq), and you have generally week fold changes and moderate alterations in your condition, you could try in R a "gene-set" approach, using approaches like roast, mroast, camera, included in the R package limma.
Or if you want a "network-based" analysis, you could try the R package SPIA, to infer and examine a more detailed conclusion about the direction of alteration in a specific pathway. Alternatively, you could start with a simple GO/KEGG functional enrichment analysis, using your DE list with fold changes, with a hypergeometric of Fishers exact test. Of course, there are numerous standalone tools outside R, such as Enrichr, which cover many possibilities and provide a lot of biological information.
Overall, each approach has its own advantages and disadvantages. Regarding your question about separating or not up & down regulated genes prior functional analysis, again you could try both approaches. For instance, whereas the grouping of up or down regulated genes, offers a more "direct" interpretation and "efficient" grouping of genes into common functions (especially for biological processes, etc)-in their direction of expression alteration-on the other hand, a separation of the initial list of genes, might result in a possible "loss" of interesting terms or pathways, due to the fact that genes, which could participate in a common mechanism with opposite pertubations. So again, it is up to you.
However, it is not appropriate to infer from a simple mapping of fold changes, that a pathway is up or down-regulated, as there are numerous interactions and molecular cascades. For starters, take a look at the following reviews and course material from Bioconductor, as they cover the topic in a much comprehensive way:
Perhaps, if you could provide more details of your experiment, biological questions and DE results, i could suggest some considerations on this matter.