What order do I watch the Minami-ke series? In the Minami-ke franchise, there's:


Minami-ke
Minami-ke Betsubara
Minami-ke Natsuyasumi
Minami-ke Okaeri
Minami-ke Okawari
Minami-ke Omatase; and
Minami-ke Tadaima


In what order do I watch these series and OVAs?
There are 4 TV series and 3 OVAs in Minami-ke franchise.

The TV series:


Minami-ke (Oct 8, 2007 - Dec 31, 2007)
Minami-ke: Okawari (Jan 7, 2008 - Mar 31, 2008)
Minami-ke: Okaeri (Jan 5, 2009 - Mar 30, 2009)
Minami-ke: Tadaima (Jan 6, 2013 - Mar 31, 2013)


The OVAs:


Minami-ke: Betsubara (Jun 23, 2009)
Minami-ke: Omatase (Oct 5, 2012)
Minami-ke: Natsuyasumi (Aug 6, 2013)


Timeline (OVAs are in italic):


Minami-ke (Oct 8, 2007 - Dec 31, 2007)
Minami-ke: Okawari (Jan 7, 2008 - Mar 31, 2008)
Minami-ke: Okaeri (Jan 5, 2009 - Mar 30, 2009)
Minami-ke: Betsubara (Jun 23, 2009)
Minami-ke: Omatase (Oct 5, 2012)
Minami-ke: Tadaima (Jan 6, 2013 - Mar 31, 2013)
Minami-ke: Natsuyasumi (Aug 6, 2013)


Source: http://www.myanimelist.net