Learn Russian: Рядом с беседкой под деревьями стоят новые тренажёры. - Next to the gazebo under the trees stand new exercise machines.

In Russian, the most important new information usually goes at the end of the sentence. By starting with the location (Рядом с беседкой под деревьями) and putting the subject at the end, the sentence emphasizes what is located there. In English, we often use the dummy subject "There are..." to achieve this same effect.

Both с and под take the instrumental case here (беседкой, деревьями). Do they always follow the exact same case rules?

Not quite. The prepositional phrase рядом с (next to) always requires the instrumental case. However, под (under) takes the instrumental case only to show static location (where something currently is). It takes the accusative case to show direction of motion (where something is going). Since the machines are stationary, we use the instrumental под деревьями.

How could we rewrite this sentence to practice this lesson's grammar, saying "not only next to the gazebo, but also under the trees"?

You would place the double conjunction directly before the two location phrases you want to connect: Не только рядом с беседкой, но и под деревьями стоят новые тренажёры. Notice how the prepositions are kept intact right after each part of the conjunction.

What is the nominative singular of деревьями, and why does it have a soft sign here?