Only use "nowadays" when talking about something that has changed from the way it was in the past. For example: "I use to eat meat but nowadays I only eat vegetables." Or " Nowadays everyone sends text messages instead of calling."

It would be WRONG to say <em>"</em>I went to the store <em>nowadays</em><em> </em>to buy some sugar." It would be CORRECT to say: "I recently went to the store to buy some sugar."

Lately and recently are interchangeable and both mean "only a little while ago I started" and are often used with the present perfect tenses.

I have been improving my French recently/lately. = Only a little while ago I started to improve my French...and I am still doing that. 

They have recently moved to Japan. 

We have been watching a lot of TV lately. 

Dion is exactly right about nowadays. That is only for when habits/customs have changed from how they were in the past. This word is used with the present simple. 

We used to walk but nowadays we take the train.

In the past, people ate dinner together, but nowadays they don't. 

I haven’t read the other comments, so I’m just going off my own thoughts.  Nowadays calls to mind the contemporary, like what’s happening now, not 10 years ago. Nowadays cell phones are more sophisticated than they were 17 years ago, as an example. Lately, seems to be something I think of as being even more immediate in time. Example, „ Nowadays our internet connections are faster than they were 17 years ago, and mine normallly is good. But, lately my internet connection has been poor. Not sure what’s wrong with it.“ 

Nowadays is more general contemporary time reference, while lately is much more immediate, like of late or over the last few weeks (not years) ago.  

Lately refers to something that may still be ongoing, but started in the past. "Lately, I've been living in Japan."

Nowadays refers to present time, which changes the construction of any statement containing it from past to present. "Nowadays, I live in Japan."

It basically means "currently," but with a broader sense of time (i.e. days, months, or even years).

Currently is more like "at the moment" than "nowadays."

A common construction used in its place by many is "these days." I'd argue it's more common, these days

Ironic that I ended that statement with it. Nowadays sounds quite vestigial. (Like something my grandparents used often.) I'm 100% sure I still use it, and may even alternate between the the same dialog during conversation. I tend to do that when I hear myself using the same work often (switching to synonyms).

I really appreciate all of you.