The Saxony Switzerland is best accessible from Dresden by suburban train (S-Bahn) or by car.
Getting to Saxon Switzerland by train:
Good and comfortable. Even faster than car due to a direct line along the river Elbe. Look for train line S1, direction "Bad Schandau". The closest station is located in the town of Rathen, it is then a ferry ride across the Elbe River to start of the walking track up to the main lookout location.
Getting to the Saxon Switzerland by car:
It may be a little tricky to find the roads out of the city Dresden if you're not used to. Follow the signs to Czech Republic (Decin) or Bad Schandau. Be aware of traffic jams at hot summer weekends. A good bench mark is the Elbe River. Simply follow it to the east.
Getting to Saxon Switzerland by boat:
If you have some time, try this. Old beautiful steam ships (there are modern diesel too, so take care when you buy tickets) bring you up the river through the beautiful Elbtal. Relaxing trip. You will have a good look to the engines! Before 1990 they were fired with coal, now oil is used, but they still have steam engines.