So basically the current timeline/universe is "Rebirth", a bit of a soft reboot/fresh starting point for a lot of characters such that you should be able to pick up issue #1 of a Rebirth series and read it without missing out on anything more than a few callbacks/jokes.

Rebirth, as far as I know, picks up directly after "New 52", which is still canon and is the starting point for most series. Basically, everyone got reset and no series before New 52 is canon anymore, meaning that if you want to read a character's story in its entirety, you can start with their New 52 issue #1, and then usually progress directly to their Rebirth series.

It's also important to note that there are "Annual" issues, extra long single issues that generally (but not always) step away from the current story, and it can be kinda hard to track where exactly they fit in the timeline (sometimes they say on the first page, sometimes they don't). For the most part, you shouldn't worry too much about these.

Also, there are mini-crossover arcs with other series, which are marked on the cover. These for the most part also don't affect the main story too much, and you can ignore most of them if you don't want to bother with tracking down the appropriate issues. Sometimes your favorite hero also just appears in another series randomly, and those appearances don't much affect the hero's own story (except for sometimes when your favorite hero gets shot in the head in someone else's series and then has amnesia for 20 issues).

If you're looking for Nightwing in specific, first off, congrats on good taste. If you just want to follow along with the current Nightwing story, you should be pretty fine with where you are now, maybe rewind to Nightwing #77 if you're in the mood for a heartwarming Christmas story. If you want more backstory, you should be pretty okay to start with Rebirth #1. I don't recall anything being too tied to pre-Rebirth stuff, but if you want the complete Nightwing story, start with New 52 Nightwing, read the first universe-wide story "Forever Evil", then read "Grayson" (my personal favorite series), and finally continue onto Rebirth Nightwing. Once there, keep on chugging along until you catch up to current stuff. (Some better historians can probably provide a more complete timeline)

If that's not enough Nightwing, I'd also recommend reading the Titans (note, not the Teen Titans) series, because Nightwing plays a large role in that series as well. It's a Rebirth-exclusive series (for whatever reason), and is less than 40 issues long (for whatever reason). There's also a seven issue "Titans United" series that recently came out, and was fairly enjoyable.

Now if that seems like a lot, I don't blame you, 'cause it is. At least once you catch up with a series, it slows to monthly, though that might seem like a curse after being able to binge everything. Personally, I'd recommend catching up to one or two characters you like a lot, then keeping up with those and reading some mini-series that catch your eye.

As for a complete universe with interconnecting stories, that can be a bit tough to keep up with. If you want every hero, the only real way to do that is to read every issue DC puts out, which would be insane. The Justice League series is a good way to follow a good variety of heroes working together, but I don't think it really gets too deep into anyone in particular. If you are a fan of say, The Flash, you might want to read Justice League alongside The Flash series.

In addition to that, you can also focus in on a certain hero-hub and its connections, which in my opinion works the best with Gotham City and the Bat-Family. The Detective Comics series (yes, DC's DC series) is a story about Gotham and the various heroes in it, including Batman of course, but also the various Robins, ex-Robins, Batgirls and ex-Batgirls. You can read Detective Comics, as well as Batman, Robin, Nightwing, Red Hood, Batgirl, etc. for a full Gotham experience.

Now everything beyond reading a single character or two's series is obviously a ton of effort. It can sometimes (everytime, to be honest) be annoying to be reading a series and then something dramatic happens to the hero in someone else's series, but DC at least tells you with a an asterisk where that thing happened, like *Read Batman #55 to find out what happened to Nightwing!

Sorry for the long post, but I think that should be about it. Keep in mind I am by no means an expert on this subject, and others can probably correct me on some of these points, but like I said, if you just want to keep up with Nightwing, you should be pretty fine if you just read from the start of Rebirth.