The Dubai airports advice on transferring between terminals is:
Moving between terminals
When connecting between A, B or C Gates, all the gates are within walking distance. The only exceptions are the A Gates. For these you take the airport train.
If you are connecting to or from F Gates (located across the tarmac), your airline will tell you what to do. Services include free shuttle buses, Marhaba service and public transport.
Terminal 2, the terminal you wish to depart from, is F gates, so the above advice applies.
The same page also notes the transfer desks for each terminal:
Terminal 3 (A, B and C Gates)
Transfer desks G, H and J are for all passengers
The shuttles run regularly, but not on any given schedule - basically one is always waiting and departs when the next one arrives.
You may want to look into using Marhaba Services, who will meet and greet you at T3 and transfer you to T2 along with your luggage. They even have a service where they can collect your luggage for you.
Qantas also advise that you can use public transport to transfer between terminals:
Bus stations are situated opposite both Terminal 1, 2, and 3. Local buses 4, 11, 15, 33 and 44 can be used to connect with Terminal 1 and 3, bus 2 connects with Terminal 2.