oh sorry - didn't see your post - you're in the blue start.
Best route (depending on where you hotel is and which tube is closest) is:
going to Barbican station. Getting circle line (going east) one stop to Moorgate.
Changing at Moorgate to the Northern Line (going south) to London Bridge
Then at London Bridge go upstairs to the rail station and get a train heading to Blackheath.
Shouldn't take you long.
www.tfl.gov.uk This website will be able to tell you everything you need.