With remote flash, groups are used to have separate power settings for each group (e.g., Group A shoots at full power, while Group B shoots at 1/2 power, to have 1:2 ratios between your key and fill, etc.), not for separate firing. To switch which lights are firing, you have to manually turn groups on and off between shots.

I'd recommend using the YN-560-TX on-camera on the hotshoe, because it has the easiest interface to manipulate the group settings. As you cycle through the Mode for each group with the MODE button, it will go through M, MULTI, and --. -- means the group is turned off. To alternate between group A and B, you'd have to set Group A to M and Group B to -- for the first shot, and then Group A to -- and Group B to M for the second.

You can use the cable with the YN-560-TX on the camera's hotshoe, and then use the RF-605 in-hand as your remote, if you don't want to use the camera shutter button. The YN-560-TX can be a receiver for the shutter, while simultaneously being a transmitter for the flashes.

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Addendum: The RF-605 should work the way you're envisioning: i.e., if the icon is on/off for the group on the TX-mode 605 on camera, then only those groups should fire, however, in a Flickr discussion thread, someone posted an email from Yongnuo where they state the grouping function on early copies (pre-Nov 2014) of the triggers doesn't work.