In New York City, my wife and I, along with others, are looking to
purchase a building. The others are basically two other groups, each
group being a couple or individual.

The idea being, if we purchased a 3 floor building , each group would
have one apartment and access to common areas
(roof/garden/hallways/basement etc.). Total costs and legal fees would
be divided equally among the group.

I imagine this is very similar to a coop (if not the same), where we
have an legal agreement regarding maintaining common areas,
maintenance, heating, and sub-letting, eviction etc. For the sake of
argument we're also assuming the building will not require major work
or renovation.

Financially I think its fair to say that we all have good or excellent
credit but not much in the way of cash savings, and each "group" earns
between 50K and 70K per year. We are all in our 20s and 30s and are
first time buyers.

My question is basically; what is the best way to go about this?

We're just looking into this now so apologies if the question appears
to be very broad, it is. Please let me know if you need clarification
and I'll do my best.

Wow, I thought you would never ask.  I have been trying to do that
since I graduated from college with people that were sensible.  The
two best agreements are a tenancy in commons agreement.  Although,
from my real estate classes in undergrad, my professor told me that it
is harder to sell.  However, if you keep enough cash to buy everyone
out, it is not a problem.  The next best agreement is called a LLP. 
In this format, all of the partners are limited to what they have put
in, or their payment.  This could be a littler harder to get people to
work together on. I would love to get a piece of New York, in a joint
deal.  However, one of the companies I was researching and considered
acquiring was wellington.  This is precisely what needs to happen to
go from partnership to reit to the big board. If you have problems
with the paperwork, I suggest buying a reit, first and looking at how
they do their books.  Congratulations,  Excellent idea, and if you
need further advice or assistance let me know. The exciting part is
that once you have that money in a pool, you can really make some
moves with partners who have a vested stake in something meaningful.
  
http://www.wellsref.com/index.html
profiles.yahoo.com/testifier

Also If you would like one of my spreadsheets that calculates debt,
each partners liability, appreciation, and so forth; let me know.