Q:

Does the crew conduct any actual exploration of Atlantis?

I just started watching Stargate Atlantis series (I am on episode 10 of the first season) and while I like each episode I do not understand why they are not exploring the city itself. Atlantis was the whole point of their journey and now all I can see is them travelling here and there without actually studying the city. I mean... that is THE Atlantis! I cannot stress this enough - THE Atlantis.
So the question is - is there going to be more exploration in future episodes or should I expect just more of those exploration missions via the stargate?

A:

Yes, there will be more exploration episodes, or at least ones containing events resulting from exploration.
These include, but are not limited to (spoiler alert!):

 The discoveries of various labs, including a secret lab - unknown even to the original inhabitants. There's also a virus outbreak, ascension machine, additional jumper bay, underwater drilling platform (near Atlantis), and others.

So there'll definitely be more Atlantis-oriented episodes.

A:

Yes, there will be some exploration of the city. The reason they aren't exploring yet is due to lack of power. They first search for a new power supply before opening up new areas of the city. 
I will warn you, there isn't much city exploring and not many (if any) focused purely on it.

