Examine why the ten young storytellers bring their servants with them to the countryside. Why do they do this? How do the storytellers achieve their goal, or do they?

The servants accompany the young storytellers to the country so that someone can deal with the unpleasant business of everyday life. They need people to clean, to procure goods, and to take care of them while they tell their stories. The storytellers are wealthy and privileged people who aren't used to taking care of themselves. Also, by sending the servants out, they're able to isolate themselves from updates on The Black Plague.
Many, many people in Florence were already dead when the group set out. Boccaccio says:

"How many grand palaces, how many stately homes, how many splendid residences, once full of retainers, of lords, of ladies, were now left desolate of all, even to the meanest servant!"

Since many people in the servant class likely didn't have second residences to go to or the option to leave the city, the trip was likely also a positive thing for the servants. Even though they had to continue to work and to acquire things for the storytellers, they weren't surrounded by the effects of the Plague in Florence.
The storytellers do achieve their goal. During their two weeks at the villa, they're able to forget the horror of the illness that's tearing through the city. They're able to have a pleasant moment in the face of the death of so many. They are also able to stay away from updates about what's happening, as they hoped—because of the servants who act as go-betweens.

The storytellers are members of the Florentine social elite. The very idea of going anywhere without their servants would've been unthinkable to them. These young aristocrats have escaped to a country villa to escape the ravages of the Black Death. And although we might think they'd have more pressing concerns on their mind, they're nonetheless determined to maintain due propriety. Whether fleeing the plague or not, Florentine nobles must always have their servants about them. It's simply the done thing.
As to the storytellers' goals, they simply want to amuse themselves to take their mind off the deadly plague raging outside. In telling their tales, these young aristocrats are affirming their humanity in the midst of such unimaginable death, disease, and suffering. The Black Death may be cutting a huge swathe through the population of Europe, but eventually it will pass. In the meantime, life must somehow go on, and from a psychological standpoint, it's important to try and act as normally as possible to keep your spirits up. That's what the storytellers try to do in The Decameron, and it would seem they are largely successful in their endeavor.

The ten storytellers bring their servants with them in order to serve their needs in the countryside.
Essentially, the servants are the last surviving employees in their masters' households. Since none of the storytellers want to stay at their former residences, they decide to head to a palatial residence at the top of a hill, just outside Florence.
The servants play the role of protectors along the journey. Then, at the palace, they return to their regular roles. It is Pampinea who gives the servants their jobs upon arrival. She makes Parmeno (Dioneo's servant) the steward of the house during their stay. Parmeno is picked to oversee the needs of the entire household.
Meanwhile, Sirisco (Pamfilo's servant) is to be "purveyor and treasurer." Basically, Sirisco will purchase the daily necessities of the household and handle the financial responsibilities that correspond to this position. He is to answer to Parmeno, of course.
Tindaro, another servant, is to tend to the needs of the three male travelers (Filostrato, Panfilo, and Dioneo). For duties in the kitchen, Pampinea appoints Misia (her own maidservant) and Licisca (Filomena's maidservant). Meanwhile, Chimera (Lauretta's maidservant) and Stratilia (Fiammetta's maidservant) will focus on keeping the ladies' chambers clean.
So, we can see that the main duty of the servants is to ensure the smooth running of the household. They are to see to the comforts of the travelers. The servants have been told that they can leave the residence to run errands (such as purchasing food), but they cannot bring back news about the plague from without. Basically, the storytellers want to insulate themselves from tragedy and suffering for a time.
In many respects, the storytellers do achieve their goal. They tell stories to amuse themselves and are largely successful in forgetting about the plague for two weeks.