A "family tree" (a chart showing all direct ancestors of a single person) is rather simple and easy to follow.
A "family group sheet" (a list of parents and siblings, with spouses, of a single family unit) is also fairly simple and easy to follow.
Combining these two things for a single family (each node of the family tree needs its own family group sheet) can get rather complex.
Setting up a combination of multiple families with all the links between them over a lengthy period of time quickly becomes very complex.
A "single view" for such an arrangement, with all the crosslinks, would be completely illegible.
The best compromise for this would be to develop a chart for each family, with spouses. Each spouse would have their origin family prominently marked on the chart.
Flowchart software would probably be the best thing to use here. As Michael mentioned, however, it would require more than one view per family, as it would indeed get highly complex. Even FamilySearch.org, a high-end genealogy service, doesn't really let you see the whole "family network" at once. However, an interesting idea is that it lets you focus on a single person somewhere along the line and see both their ancestors and descendants. So perhaps you could choose an individual and focus some trees around him or her.
As for the recommendation of software, a quick Google search for flowchart software pulled up many results. A lot of the results are proprietary, but things like Gliffy and SmartDraw should work — Gliffy will give you 5 free flowcharts (and it's pretty easy — I was able to make a very simple two-generation chart in less than a minute on my first try), which may or may not be enough depending on how complex you wish to get, and as for SmartDraw, it does have some purchasing options, but also a free download. If you dig deeper, you might find something else that suits your needs better.
You might also want to try some wiki-like software if the flowcharts don't work out for you or if you want some extended note-taking capability.
Mind mapping software is designed to link unstructured but related information, which sounds like what your families will be like. You might want to look at something like MindMeister.
May I suggest the use of Gramps. It is open source freeware and specifically developed for genealogy. It can display really huge and complex charts. I use it for my world building to keep track of family relations. Creating a custom calendar is a bit of a hassle, but possible. Gramps has aso lots of other features, you might find helpful. You can also look through the recommendations of this site. ScionPC for example can easily create a custom calendar, though display options seem to be more limited compared to Gramps. So far I only used Gramps and I am entirely satisfied with it.
If you have access to a Macintosh there is a great geneology app called MacFamilyTree by Synium Software. It can do everything you asked for.
A suummary of features is a good idea (from the Synium site) :
Explore your personal family history and discover your origins, your ancestors and how your family has evolved over the course of time. MacFamilyTree 8 offers numerous options for you to capture and visualize your family history. No matter how you want to illustrate your family relationships ... Conduct your research with the help of a free online genealogy archive with billions of records, create websites or do your research on the go with MobileFamilyTree for iPhone, iPad and iPod touch. MacFamilyTree also lets you create family books in a flash - share them as a PDF or print them.
Personally, I'm a fan of FamilyEcho. It's simple, easy to use, and free.
It also lets you make multiple family trees, add information about each person, set a person's gender as nonbinary (though it isn't labelled as such), and add gay relationships with children.
The only problems that I've found are that it doesn't let you give someone multiple partners at a time (it sets ones you already added as ex-partners) and there are no way to set more than two people as parents. The latter one makes it difficult when trying to map non-nuclear family units or trying to add an adopted person's biological lineage.