Doesn't the form "ar" represent third person in plural? Isn't the "am" correct for first person plural?
No, for this purpose it is as follows:
- I would like to = mi-ar plăcea să
- You (singular) would like to = ti-ar plăcea să
- He/she would like to = i-ar plăcea
- We would like to = ne-ar plăcea să
- You (plural) would like to = v-ar plăcea să
- They would like to = le-ar plăcea să
Think of the verb as the equivalent of the English verb "please". That way the sentence could be translated as "Going shopping would please us".
You can think of it like this: "Would going to the sea be something to your liking." "you" is not active in this sentence. Same in german: "Wuerde es dir gefallen,..."