No one is allowed to just show up at base camp and expect to be able to advance to any of the other camps.
First of all, there is a lot of paperwork that must be completed before you're even granted a permit to climb Everest, which includes a letter of recommendation from a national climbing association. The permit to basecamp is only $50.00, but if you want to go any further, then you need to purchase an $11,000.00 permit. If you get caught above basecamp without a permit, the fine is $22,000.00, and you might even get jail time on top of that.
Second, the route from basecamp to camp 1 is the traverse through the Khumbu Icefall, which isn't just a walk in the park. It's a treacherous traverse over deep crevasses and under looming seracs, which requires the use of ice axes and crampons, and the only way through is over ladder bridges and with the use of fixed ropes. Navigating the Khumbu Icefall is a little more technical now than it has been in past years, because they changed the route in order to avoid the danger of avalanches. Many people have died on Everest between basecamp and camp 1 by being swept off by an avalanche into a crevasse.
Image from National Geographic
Expeditions pay big money to have all those ladders and ropes packed all the way up to Everest, and to have experienced Sherpas fix them along the best route, which changes every year as the ice continuously moves and changes form. They aren't going to simply let you make use of their equipment. One of the biggest hurdles on Everest is getting around all the officials who want a piece of the tourism revenue pie. Tourism is number one industry in Nepal, and climbing in the Himalayas represents a large portion of that industry.
If you want to check out any of the camps above base camp, you're going to have to bring some deep pockets with you. I'm not going to say it's not possible, but if it is, it's not going to be cheap.
After finding a fee schedule, it looks like the $11,000.00 fee is only for Everest Summit bids in Spring. In Autumn it appears you could go as high as Camp III for only $900.00 if you purchase a Lhotse summit permit (Lhotse is the peak right beside Everest, you climb the Lhotse Ice Face to get to Camp III, then head South up to the Summit of Everest, or North to the Summit of Lhotse).
Either way, you're going to need a permit, and likely a guide. Your best course of action is to hook up with a expedition like the one linked above. Expeditions help you cut through all the bureaucracy by dealing with all the government officials on your behalf. It's still going to cost you something though, and you're going to need to buy all your gear, which you may want to do in Nepal, I've heard you can get great deals on stuff if you buy it there as lots of climbers leave their used gear behind after their once in a lifetime summit attempt.