Stormy, thanks for the best answer star. But, shoot; I never came back and answered your follow-up question. And nothing close to awe is warranted. It's more like having a really big jigsaw puzzle and, after almost 18 years, you finally have the corner pieces and most of the pieces on each side. It's tedious, not impressive...and we were kinda slow on the uptake.
No special tools are required. Uptopic were mentioned network phylogram utilities like SAPP and Fluxus and Gephi. These can be quite useful, but they're only a piece of the aforementioned puzzle. And while they serve as good indicators, they can't be fully relied upon. Utilities like that don't know about actual mutation rates or the exigencies involved (like STR back-mutation or SNPs in palindromic regions where the precise position may be ambiguous...this is one reason the Big Y test skips such a large percentage of the chromosome).
I'm pretty much inactive on Facebook, but I did join a private group a couple of years ago that may be of interest to you regarding phylograms and genetic networking: https://www.facebook.com/groups/NetworkGraphsForGeneticGenealogy
The two requisites for your own yDNA project are having the highest quality genealogies possible of the participants (paper trail rules!), and the full set of derived (positive) SNPs and all tested STR values. For the latter, it definitely helps if a group project admin at FTDNA is involved, and if all participants set their rights to at least "typical" and to allow data to be published on the group project page. FTDNA will update test takers' results based on new discoveries (novel variants moved to named SNPs), new alignments to the haplotree (new branches created or branches realigned), and of course new matching reports. The rights-permitted admin can see all of that.
Barring that, each participant involved would need to download all their data (SNPs, STRs, matches...and of course family trees) and send them to you. Then you just start reconstructing that jigsaw puzzle!
Since you aren't R-P312 upstream, unfortunately you can't make use of Alex Williamson's "Big Tree" (uploading to YFull is an option, though). But that's where FTDNA recently adopted the look and feel for the "Block Tree" it displays in the Big Y results area. I'm really glad they did that because it's a good go-by and FTDNA keeps it up-to-date with the full haplotree. You can see my little area of the Big Tree here. We have 19 Big Y kits uploaded there and about half-again as many in the subproject that aren't. But you can see how it lines up with that big JPG I linked to uptopic, how the branched SNPs descend hierarchically (with each bounding box as a hierarchical step) and where bifurcation--splitting--occurs. In your own private project records, of course, you'll also be keeping track of individuals' novel variants because new tests matching those could pop up at any time.
A quick word here: if you're dealing exclusively with Big Y results, you're basically golden because you're matching apples to apples. If (as I hope but incorrectly predicted a couple of years ago) that whole genome sequencing really starts to take hold for genealogy, you'll have to get more sophisticated. Getting the Y chromosome's SNP data out in a VCF (variable control format) file isn't all that difficult, but that shows you only the SNPs that differ from the in-use genome reference map, so you likely will have to do some fill-in-the-blank work of your own to get the full picture. And WGS tests show data from all along the chromosome, even in the area that FTDNA doesn't test because it's so full of repetitive and indeterminate information. Also, you'll be dealing with a rather cryptic looking format when you check the VCF files, and the results will show only the loci position information on the genome map in use plus the alleles...so you'll have to cross-reference that with SNP names and values that FTDNA uses. Manually checking STRs within the data (those are in the BAM and FASTA files, not the VCFs) is, for now, mostly out of the question.
Okay. Sorry; too much information. Net message: just be aware that if someone says they'll do a WGS rather than the Big Y, be happy...but know that you won't be able to easily or immediately incorporate that data into your project. For the near future, for the Y chromosome, Big Y rules.
Then it's just a matter of working the jigsaw puzzle backward. John, Joe, and Jim Smith all descend from Hezekiah Smith, one of four sons of Absalom. All three test. You take their in-common data and align them to Hezekiah in the genetic tree you're creating. If John isn't a good match with Joe but is with Jim, then there may be a problem with Joe. At some point--before or after Hezekiah and Absalom--you'll see consistencies in both SNPs and STRs. The consistencies allow you to spot and catalog divergence. After a while it also becomes relatively easy to tell where different surnames begin to fit, and to make reasonable determinations about whether there is a suspected non-paternal event or if you've cataloged a surname simply adopted a long time ago by that patrilineal branch.
Dating the hierarchy is still more art than science. FTDNA had told group project admins almost a year ago that they were plans to start including SNP dating estimates with the Big Y results in our individual control panels. Hasn't happened yet. A lot of folks use the dates as calculated at YFull and displayed on their haplotree (which is undergoing a version update right now; still available, but look for revised information to propagate shortly after the first of the year). Can be a place to start. But if you dig a bit, you'll see that their estimates are based on a paper the YFull folks self-published at ResearchGate; it isn't peer reviewed or validated. Still, it doesn't seem their dates are too extreme; the rumor was that FTDNA's dating wouldn't be factors of magnitude away from what YFull is doing.
That said, the more full-sequence data you gather within your own project, the more accurate near-generation information will become. For example, say John, Joe, and Jim all match nicely under Hezekiah, but you have an additional SNP defined for two descendants of Hezekiah's brother. As the data come in, you can start to confidently note that the split happened no earlier than Hezekiah's father, Absalom. Voila! A pretty precise bifurcation date.
It's a great time to dive into such a project. The data are available now. I joke that we were kinda dense because it took us over 17 years to get to where we are in our Williams subproject, but it truly wasn't until the appearance of the Big Y test circa 2014 that we had more than piecemeal STR information to work with. It's been leaps and bounds since then.