We've had related questions from prospective converts and one that asks specifically how to behave in a particular group's shul. This answer will overlap with answers there, but I'm going to focus specifically on the "Christian visitor" aspect of your question, because even though prospective converts can visit, you might not automatically generalize that to people with no interest in conversion.
The general answer to your question is yes, you can attend services. Nobody's checking ID at the door,1 and the presumption in the vast majority of communities (perhaps all of them) is that people who show up belong there. As discussed in the linked questions, dress appropriately and make sure you don't do things in the service that only Jews are allowed to do; decline any honors you're offered, and if you're a man and there aren't at least a dozen men there, you should be concerned about being accidentally counted in the minyan (quorum).
As a Christian you should be careful of some additional things. Please don't wear a visible cross; it's not that it's forbidden, exactly, but it's going to be seen as kind of rude by some portion of the people there, because it's a symbol that has been used to oppress Jews for centuries. It's not equivalent to a Jewish man wearing a kippah or a Muslim woman wearing a hijab, both of which are signs of the person's religious identity; the cross is more powerful, more "assertive". If it's something you wear all the time for your own religious reasons, try to tuck it under your shirt or something.
Similarly, you probably want to avoid talking about Jesus, and if you do talk about him you shouldn't refer to him as a rabbi (let alone as divine).
On the other hand, saying that you're a student visiting from such-and-such church is fine. I live in a city with several universities; we get visitors doing research for classes sometimes. You don't have to hide your identity. In some synagogues, introducing yourself that way will get you an offer of somebody to sit with you and help you navigate the prayer book.
See also this general overview of what happens at Jewish services and how to behave.
1 One exception: on the high holy days (Rosh Hashana and Yom Kippur), seating is often very limited and thus restricted to members of the synagogue. Don't go then. Besides, if you do you're signing up for a 4+-hour service, which probably isn't what you wanted.