chigwa-e gassdaga miyongsil yeyakdo haesseoyo.

Both, depending on how you translate it.

• 치과 literally refers to dentistry / a dental clinic
• But in everyday Korean, 치과에 가다 is the normal way to say go to the dentist

So in this sentence, English would usually say the dentist, even though the Korean word is closer to dental clinic.

에 marks a destination here.

• 치과에 가다 = to go to the dentist / dental clinic

With movement verbs like 가다 (to go), 에 commonly shows where someone is going.

Here, 갔다가 means something like:

• went and then...
• after going...
• stopped by and then...

So 치과에 갔다가 미용실 예약도 했어요 means the speaker first went to the dentist, and then did another action: making a salon reservation.

It connects two actions in sequence.