Why 了 is not used, and what is the meaning of 玩 in 上个星期天，我跟好朋友一起去林学院的农场玩?
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.