While a DSG is very similar in operation to a slushbox, there are a few differences. 

First, always set the parking brake before you put it in park (you should always do this on any VW/Audi since it sets the rear brake caliper self ratcheting mechanism and avoids stressing the parking gear pawl/shifter). 

Always press on the brakes before starting the engine (you should already be doing this with any car).  The car could slightly lurch forward if you start the engine without the parking brake or brakes on.  This condition is most likely under cold temperatures due to the viscosity of the DSG fluid transferring some motion from the engine to the transmission. 

Unlike a slushbox, if you take your foot off the brake, the car will not inch forward.  When coasting with your foot off the gas you'll experience engine braking, just like a manual transmission.  However, the transmission will downshift as you slow down.  With a manual transmission, I prefer to do a combination of downshifting and neutral depending on the conditions.  When stopped on a hill, the car may roll back a little if you don't apply throttle, just like a manual transmission.  Most later DSG have a hill holder which holds the brakes until you step on the gas.

The dual clutch system tries to predict your shifting but it is not adaptive off driving style.  It will adapt to adjust for internal wear to keep the behavior of the transmission consistent.  This is verified by DSG transmission tuners.

To maximize fuel economy, shift the DSG manually.  There are other advantages to a DSG transmission so don't feel like you are "wasting" the additional cost of the DSG by shifting manually.  Drivers trying to maximize fuel economy report the average DSG mileage about 3-5 mpg average lower than a manual transmission.  As an example, if you drive 12,000 miles/year and diesel is $4/gallon, this is only a difference of about $12/month, so don't be discouraged by the small difference.  Although some of this difference can be attributed to build variation between cars, driving style, and fuel quality, the general opinion is that DSG will give slightly lower mileage than a conventional manual, everything else being equal.