When you plug your phone in to your PC, you can copy media across using Windows Media Player; you no longer need to use the Zune application on the desktop. You can also just copy the files across using Windows Explorer as it will show up under "My Computer".
On the phone itself, the app is now called "Music+Videos" and is "X-Box green", with a headphones/...
The last version of WP you can install on your Lumia 610 is Windows Phone 7.8.
The exact version is 7.10.8862.144, released the 14th March 2013, as you can see in the WP 7.8 versions history table.
Assuming the download doesn't take too long, the update should take 30-40 minutes.
Zune isn't supported in Windows Phone 8, have to use something else to synch music:
Zune Music and Groove Music are the same in Windows 10 desktop. So try running
in a command line prompt or Power shell with admin rights.
There is no way to download, backup, modify or work with SMS / texts in any way using the Zune software. Zune can only be used to move music, pictures, podcasts and videos to Windows Phone 7.X devices.
There's also no other software that would allow such kind of access as Microsoft has not yet opened up any APIs for developers to use.
I had the same problem running my Nokia Lumia 610 connected to a Windows XP computer, I tried turning off the Windows Firewall located in the Control Panel and it did not work. So I turned the firewall back on and added Zune to the Exceptions list of programs in Windows Firewall. Next I tried the other recomendation mentioned by tomekpe by turning on ...
I had experienced the same issue, and after researching on this page http://support.xbox.com/en-AU/xbox-on-other-devices/windows-phone-7/wireless-sync
you can see on section 'Compatible wireless networks' that states 'Zune players do not support the 802.11n standard.'
Since my network is 801.11n, I guess I won't be able to sync my phone...