General Tips
(1) [MOUSEOVER TOOLTIPS] Take the time to read the help text from the mouseover tool tips. There are a lot of tips in there (since they can easily and quickly be changed via the text file) that have continually been updated with each subsequent patch that you probably (won't) find in the manual.
(2) [CUSTOM LAYOUTS] Known them, love them, enjoy them

my favorites for max information on what I need to know to make decisions: SHIPS LIST for moving ships (picture - only for ability to reduce row height and add more rows per screen, Name, Movement, Damage, Supplies, Fleet Name, Status Icons); PLANETS LIST to send colonizers (Name, Size, Type, Atmosphere, Values, and Time to Arrival); CONSTRUCTION QUEUES LIST for updating queues (Picture, Name, Colony Type, Items under Construction, Time to Completion, Status Icons); COLONIES LIST to check cargo/facility status on change colony type as needed (Picture, Name, Colony Type, Facilities (current & total), Cargo (current & total) and Status Icons.
(3) [MOVING] Hotkey "M" for moving - lifesaver each turn
(4) [PRIMARY SCREEN HOTKEYS] Get to know the major hotkeys for primary subscreens you use each turn (Log = SHIFT + L, Research = SHIFT + R, Ships = SHIFT + S, Queues = SHIFT + Q, Colonization = SHIFT + P, Designs = SHIFT + D, Intel = SHIFT + I, Diplomacy/Empires = SHIFT + E and Finances/Overview = SHIFT + O)
(5) [END OF TURN DIALOG] Leave the end of turn warning dialog box on - that way it will warn you when you have unallocated points, ships without move orders and unanswered messages
(6) [SYSTEM DISPLAY] Leave in empty queue notice, but turn off the full queue (no need to have them both). This way the empty box will appear on the system display when you have an empty queue. Also, I like to turn off WPs since they are on every planet, but keep on others that work into my logistics system (sats, troops, fighters, mines).
(7) [CYCLE ALL SHIPS EACH TURN] Use the spacebar to cycle through ships without orders each turn. It will quickly select that ship so you can give it orders.
(8) [REMOTE LOAD/DROP] Key hotkeys for your logistics system. I find the defaults hard to remember, re-designate to what you like and use.
(9) [RETROFIT, SURVEY, QUEUE, COLONY TYPE] More hotkeys that I use regularly "CTRL + E", "/", "Q", ";" respectively.
(10) [WAYPOINTS] Currently designated with CTRL + #, and send ships to by simply pressing the #. No current way to rename them, but its on our list
(11) [SPOUSAL TIME ALERT] Clicking on the header bar over the star date will reveal whether its time to attend to your spouse or kids. Also, clicking on the resources will cycle the intel/research points and your normal resources.
Hmm, that's it for now.