Xbox – Capturing Voice Chat on Xbox One & PS4

To use the Kinect microphone to chat while playing games, press the Xbox button and choose "Settings" followed by "Kinect".

Ensure that the Kinect sensor is enabled and that it can hear you (if your Kinect can't hear you, select "Kinect doesn't hear me" and go through the audio calibration process).

Select "Chat Microphone" and turn this to "on". This will default to using the Kinect microphone in game.

If you need to mute the Kinect microphone to send a voice command to your Xbox, you can do so by hovering your hand over the speech icon in the lower right corner of the screen.

No, you can't voice chat between Xbox One and 360. There was a Q&A session with Microsoft on IGN in August when this was confirmed:

Is there cross-console Party Chat? i.e. since it’s all “Xbox Live,” can people on Xbox 360s Party Chat with their friends on Xbox One?

WHITTEN: For a variety of technical reasons – including the sound quality that I talk about below – we don’t support voice chat between the two systems. You will still be able to send and receive text messages to friends on both systems, however.

This doesn't say anything about joining parties across consoles, however given that cross-console voice chat doesn't work I wouldn't expect to be able to join parties across consoles either.

You can send text messages across consoles, however.