Clean the battery terminals (posts and clamps).
Dash lights require very little current.
Starter requires huge amount of current.

Does the starting problem occur when it's hot outside?
if so, it could be the main relay has gone bad (cold crack soldier joints in the relay itself).

Check all the grounds (and clean if necessary)
-transmission
-engine at valve cover
-thermostat housing (this is the main ECU ground)
-car body ground
-negative battery terminal

Hard shifting into third or any other gear means the syncro for that gear is worn and might be going bad. (nothing electrical there)