A lot of questions, will try to reply to all
1. If you wish to explore Old Town Dubrovnik there are public buses, from the Main Bus Station in Dubrovnik bus numbers 1A, 1B or 1C will take you to Pile Gate, entrance to Old Town.
2. If you arrive at the Main Bus Station there is a luggage storage nearby at Dubrovnik Travel Corner, http://www.dubrovnik-travelcorner.com/home/location-in-dubrovnik
. The same bus numbers from above pass just 20 meters from there.
3. From Dubrovnik to Korčula you can take the bus (much faster) or the ferry. Bus tickets can be bought in advance, so when you arrive get the ticket, store your luggage at the Dubrovnik Travel Corner and you are all set to explore Old Town Dubrovnik.
Timetable for the bus from Dubrovnik to Korčula: http://www.dubrovnik-online.net/english/transportation_coach.php
Not many tourists know, but there is also a private company from Korčula that operates a bus line from Dubrovnik to Korčula, usually departing from Dubrovnik around 13.00, tickets and information can be obtained at the Dubrovnik Travel Corner.
For the ferry timetables please visit http://www.dubrovnik-online.net/english/transportation_sea.php
Korčula is also called "Small Dubrovnik", we will have a more detailed guide about this beautiful island soon.