I can't understand the role of し after 留学 in the sentence:
I heard that it is supposed to be a conjugated form of する, but I don't know a conjugation ending in just し. Any ideas?
しに行くis a way to express "go with the purpose of doing (something)".
The し is the Continuative Form (連用形) of する - you will probably recognize it as the base part of します. All verbs have a continuative form. For example, 話す would be 話し, 食べる would be 食べ, etc. Basically it's the part that comes before ます.
When you have the construction:
Verb 1 (continuative form) + に + Verb 2, it means "to do Verb 2 with the purpose of carrying out Verb 1".
So in your example sentence, 留学しに行く means "go (Verb 2 行く) with the purpose of studying abroad (Verb 1 留学する)".
But it doesn't always have to be し, as that is only the case with する. Another example phrase would be 先生に聞きに行く. This time 聞く is in the continuative form 聞き. So the phrase is translated literally as "go to the teacher with the purpose of asking something". A more natural translation would be "to go ask the teacher". You might also recognize the phrase 見に行く (to go see something). This is also the same construction
Verb 1 continuative form + に + Verb 2.