To get there from Las Vegas, take I-15 north to Highway 93 to Alamo and Ash Springs. North of Ash Springs, take the turn for Rachel onto Highway 318, then onto Highway 375. The state of Nevada, in the spirit of encouraging tourism, renamed a section of Nevada State Highway 375 near Area 51 as "The Extraterrestrial Highway." The highway signs, plastered with spaceship stickers, make great photographs.
At mile marker 34.5 on Highway 375, turn onto Groom Lake Road. The Area 51 boundary is 13 miles down Groom Lake Road. You will be able to see the warning signs, and most likely the "men in black" security guards. Do NOT pass the warning signs, or you will be arrested.
The viewpoints at White Sides and Freedom Ridge were seized by the government in 1995, and are no longer open to the public. The closest publicly-accessible viewing spot is now at Tikaboo Peak, about 26 miles east. It takes a four-wheel-drive vehicle and some rock climbing just to get there, and you'll need powerful binoculars or a spotting scope to see anything at all.
An alternative to driving all the way out to Area 51 by yourself is to take a guided tour from Las Vegas. Adventure Las Vegas offers an all-day tour that includes a visit to a ghost town, lunch at the Little A'Le'Inn and a visit to the Area 51 entrance.

