Another option to get to Altenberg is the bus. The bus line #360 links Dresden with Altenberg. Buses depart at the central railway station in Dresden and arrive at the train station in Altenberg. They run at least every hour and the ride takes about an hour.
During your stay in Altenberg buses are also the best option to get around. A major advantage is that you can start and finish your hike/cross-country skiing tour at different places. Another advantage is that you don't need to search for a parking - especially in winter! The buses serve the smaller villages (Schellerhau, Bärenfels, Oberbärenburg, Rehefeld, Neuhermsdorf, Zinnwald) in the area frequently - mostly every hour.
A plus is that the bus station in Altenberg is right by the train station and that it is protected from wind and snow/rain by large glass walls and roof.
Fare from Dresden to Altenberg is 5.10 Euro (one way). Day/week tickets available. See the vvo-website for details.
The most convenient option IMO to get to Altenberg is the regional train. You take the suburban railway S1 from Dresden (Meissen-Dresden-Pirna-Bad Schandau) to Heidenau where you change to the train via the Müglitz valley (Müglitztalbahn). Connections are guaranteed, the trains run every hour.
The train ride from Heidenau to Altenberg is quite scenic. You pass some castles (Weesenstein, Bärenstein) and cute small towns (Glashütte, Geising), before finally the train climbs up to Altenberg, offering panoramic views of the mountains. The ride from Heidenau takes about 40 minutes.
In high (winter) season direct trains run every morning and evening from Dresden to Altenberg and back (depending on good snow conditions).
Fares are 6.80 Euro from Dresden to Altenberg (one way). Day (family) tickets and weekly passes are offered (please see the VVO transportation network website for details).
Altenberg and the region are easily accessible by private car. If you like driving you can take the motorway A 4 to Dresden and then the A 17 towards Prague. Take the exit Dresden-Süd and drive the major road B 170 to Altenberg. Or stay on the A 17 and take the last exit before the Czech border (Bad Gottleuba) and drive via Geising. The latter option is a bit faster, but from there the roads to Altenberg are quite narrow and windy.
There is a big parking lot right at the skilift in Altenberg. They charge 5 Euro for a day ticket (see pics 2 and 3). Personally I prefer the parking at the big indoor pool/sanatorium Raupennest at the road to Rehefeld, just outside the town Altenberg - free and also right at the cross-country trails. Not an option for downhill skiing, though.
Please keep in mind you are driving in a mountain region. In winter you MUST expect snow and ice and difficult driving conditions. My advice is to rely on public transportation.