Which Uniden DECT 6.0 phone do you have? Uniden makes several models.
For one Uniden phone that I know (DECT-1080), first make sure that the answering system is on. If needed, at the base press Ans On/Off. From a handset, press Menu/OK and use the redial and Phone Book keys to navigate through the menu to AM Operation. Press Menu/OK and navigate to Ans Settings. Press Menu/OK and navigate to Answer On/off. Press Menu/OK and then choose On and then Menu/OK again.
To set the Out-Going Message, again go to the Ans Settings submenu at the handset, and navigate to Record OGM and press Menu OK. Select OGM 1 or OGM 2 and press Menu/OK. Record your message after the tone. To select the OGM message to be heard by incoming callers, navigate to Select OGM, press Menu/OK, select the message and press Menu/OK.
Please note that you also need to make sure that the system isn't set to Announce Only in the Message length section. Enter this submenu from the Answer Settings as above. The choices are 1 minute, 4 minutes and Announce Only.
You can also do set the OGM from a touch tone phone remotely. Press * as the OGM plays and then enter your PIN. Press 06 and then 1 or 2 to select the OGM to edit. Then press 9 to return to the main menu.
However, on the DECT-1480, you set the OGM from the handset but in a different combination. Press Messages/Mute then 5 (or wait for the announcements to end). Press 8 or press the Caller ID/right arrow to navigate through the menu to 8: Record New Greeting. Press Menu/Select. Wait until you hear "Record greeting" then speak your message normally. Press 8 to end and listen to the playback of the greeting. Press End to save the message. (Select the desired greeting by pressing 6 instead of 8.)
Most Uniden DECT phone manuals are available from the product page or this list: http://www.uniden.com/bin/venda?ex=co_disp-view&bsref=uniden&scat=dectphones_mnl
If you provide your model number in a comment, I'll be glad to give you the specific directions for setting the OGM on your phone.
I hope this helps.