Who might come along on a caravan/pilgrimage?

It depends on where & when they are going.

Middle ages, modern, ancient times? Going to the Holy Land, or a local shrine?

Are they going because of local famine & pestilence, because the pope called them to, because they hope to set up shop along the pilgrimage route?

Well, I would say most pilgrimages would have people of all class, and varying ethnicities/languages. Peasants, tradesmen, perhaps monks and priests. Wealthier pilgrims would obviously have better means of transportation and protection, with the poorer pilgrims on foot. If it's a relatively safe pilgrimage route, the size of the caravan may be enough to keep any highwaymen or bandits. Merchants and tradesmen may not be going the whole length of the pilgrimage and simply looking to find work along the way. Others may only be looking to set up shop in the destination city. Probably some form of traveling entertainers as well, hoping to make some money along the route.

Those would just be my initial thoughts.

You can look at Chaucer's Cantabury Tales for example. He wrote abotu a Knight, a Squire, a Parson, a Nun, a Wife, a Miller, etc... Those people come from all aspects of Society (Military, Religious, Artisan).

So who would go on a pilgramage? Everyone.

Then toss in the people selling food, shelter, and relics along the way.

On a Medieval pilgrimage you'd have almost everyone you could think of. Pilgrims, priests, religious orders, military orders, knights, soldiers, peasants, children, elderly... Basically everyone. Sometimes they were protected, most often than not they weren't, that is why military orders such as the Templar Knights were founded. Pilgrimages to holy cities such as Jerusalem were in large numbers so they were offered free passage wherever they went.

http://en.wikipedia.org/wiki/Way_of_St._James

Around a pilgrimage there would be merchants and other shady dealers trying to offer different items but I'm not too knowledgeable on how efficient they were.

Caravans were a different story.

Caravans were guarded, especially large ones which bore important tradable goods such as Oriental spices, velvet and other high value items. Not everyone would go with a caravan although people would often join in with a caravan because of the implicit protection and sometimes they were offered whatever was left of the food. Caravan guards were strong, experienced soldiers who were now mercenaries with a contract. The larger the caravan, the more guards you would have around it particularly since bandits were a troublesome pest.

Hope it helps.