The sentence is:
I wonder why the writer doesn't add 了 after the verb to indicate that this action took place in the past. Also, what's the use of 玩 in that sentence? And why is it placed in the end of that sentence?
Another way to see it is that 去玩 is 离合词（Separable Word). You can insert some words in between. 去玩 -> 去农场玩 -> 去林学院的农场玩. 去林学院的农场玩了 is also correct in my opinion.
去林学院的农场玩 VS 去林学院的农场玩了, the former doesn't indicate the tense and the latter indicates it happened in the past.
In your sentence, 上个星期天 already denotes when it happens, so there seems to be unnecessary to add 了 in 我跟好朋友一起去林学院的农场玩. But I can see a nuance between with and without 了.
上个星期天，我跟好朋友一起去林学院的农场玩了 sounds like the thing that the speaker is trying to tell is that he had gone to 林学院的农场 to play/have fun with my friends. Well, 上个星期天，我跟好朋友一起去林学院的农场玩 sounds like an incomplete narration, like in English: Last Sunday, while I was going to 林学院的农场 to play with my friend, ... ... Basically, I would expect to hear more context. But with 我跟好朋友一起去林学院的农场玩了, I wouldn't expect more to come (though the speaker might have more).
去了林学院的农场玩 is not correct or natural, because 去了林学院的农场 is a complete phrase. So, it's correct to say 上个星期天，我跟好朋友一起去了林学院的农场.
I try my best to expalain it with my fair English.
First, here we have two verbs “去”and “玩”. And there can be following cases for your sentense:
.....去.....玩。 In this case, both verbs are emphasized. That means, information described about these verbs are all important.
.....去了..... 玩。 In this case, with 了 after 去, the tone has slightly more emphasis on the verb related to 去.
.....去......玩了。 In this case, with 了 after 玩, the tone has slightly more emphasis on the verb related to 玩. (But this case is a bit not that natural in this sentense. Not that “fluent”.)
By the way, here 了 means already.
了 bears the meaning of what already in done in the past, but it is not a must to add in 了 in every sentence that is in past tense.
For this case, either adding 了 or not, both sentences will still have the same meaning, 了 becomes optional especially the front part of the sentence (上个星期天) already mentioned the time.
玩 has the meaning of play which doesn't make sense if translated to English. However the 玩 in this case means something like "Have fun" or "Enjoy" where the meaning is closed to "Play". So the meaning can be "My friend and I go and have fun in 林学院的农场."
了 doesn't denote 'past', it indicates 'completion of the verb'
Example: 明天完成了工作後来见我 (come see me after you finished work tomorrow)
If you add '了' after the verb '去' and wrote "上个星期天，我跟好朋友一起去(了)林学院的农场玩" that would mean you did go there, and the action of '去' (go) was from start to end, completed
Look at the following example and you can see without adding '了' after the verb, you can express action that's not completed
"上个星期天，我跟好朋友一起去林学院的农场玩, 但半路遇上交通意外, 沒去得成"
"Last Sunday, I went to the farm of Lin College with my good friends, but we encountered a traffic accident halfway and the trip was ruined
In this context, you cannot use 了 after 去, because the verb is not completed.
The original sentence didn't use 了 after 去 doesn't mean the verb is not completed. It could be either, just not stated clearly. Since the sentence end there, we can presume the verb is completed and they did went to the trip
了 is more like
Had + Past Participle in English grammar. Which indicate the action is done for and over with!
But it's not really necessary in Chinese grammar, because you already specify that the action took place in 上個星期天 (Last Sunday).
The only difference between 了 and simple
Had +pp. is that, in some cases, 了 contains some kind of
emphasize feeling. Whether the feeling is good or bad, the usage of the word make the listener think that you're profound of the experience.
If you put 了 after 去, it means that you had been to that place
and came back!
And somehow also put a emphasize on the place you go. It's very similar to the English
So it's more common to use in the situation when you go somewhere very far!
(A: I just went to Mars yesterday!)
As for why 玩 is placed at the very end of the sentence. That's because to some degree, Chinese grammar is more similar to German grammar. Which also place their verbs at the very end of the sentence.
So, if you have a little understanding of German, it will probably be easier for you to understand Chinese.