1. On the one hand they want to go abroad, but on the other hand they want to save more money.
2. On the one hand they like playing ping-pong, but on the other hand they want to go back home early.
3. On the one hand they want to play soccer, but on the other hand they want to take a rest in the shade.
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Are they all grammatical? Would you correct errors?
I'd put a comma after each instance of "hand," but all three sentences are grammatically correct.
I'd also be more likely to put a semicolon where you have the comma in the middle of each sentence -- and then I'd omit "but."
All of the sentences you provided are grammatically correct. However, there is a small issue with the phrase "on the one hand" being used repetitively in each sentence. It is more common to use this phrase once at the beginning of a statement and then use "on the other hand" to provide an alternative perspective or contrasting idea.
Here are the corrected versions of the sentences:
1. On the one hand, they want to go abroad, but on the other hand, they want to save more money.
2. On the one hand, they like playing ping-pong, but on the other hand, they want to go back home early.
3. On the one hand, they want to play soccer, but on the other hand, they want to take a rest in the shade.
By using the phrases "on the one hand" and "on the other hand" in this way, it helps to clearly express opposing thoughts or conflicting desires.