Can a gentile go to Synagogue on Shabbat and what is forbidden on Shabbat for a gentile?

Is a gentile allowed to participate in Shabbat prayers in a Synagogue (of course he/she will not count for a minyan) if they do so in order to gain knowledge of Jewish customs for the purpose of conversion?

As far as I know it is forbidden for a gentile to 'observe Shabbat' but I do not know what does it exactly mean and I would like for someone with more knowledge to clarify what is forbidden for a gentile to do on Shabbat that Jews shall do.

Also how should a gentile behave in a Synagogue, are there any special rules different than the 'normal' rules applicable to the Jews?

As a prospective convert I was told by a rabbi to start celebrate Jewish holidays and customs, but what am I forbidden from doing on Shabbat?

Answer

Gentiles can certainly attend synagogue services on Shabbat (or at any other time). I know many converts and all of them were required by their rabbis to start doing this fairly early on in the process. Conversion is in part about joining a community, so you'd better get to know it. Also, while you can practice prayers on your own, you need the experience of praying in that setting to really start to "get" those skills.

Further, sometimes non-Jewish friends of the family attend a bar mitzvah on Shabbat morning. If it's ok for them, who are there out of friendship and not out of belief, then how much the moreso should it be ok for someone who wants to join with the Jewish people?

It is forbidden for a gentile to fully observe Shabbat (see discussion here, thanks @ba). So long as there is one aspect that you violate, you're not doing that. If there's no eiruv where you live, carrying your keys in your pocket would satisfy that.

As far as behaving in a synagogue in general, decline any honors that are offered to you (like an aliyah) and you already know they can't count you in the minyan. Beyond that, you can just do what other people are doing. It it usually expected that men cover their heads; it's ok to use a kippah to do that. (Kippot are not restricted to Jews.) For more on head-covering see this question (thanks @ba again).

I wish you well on your journey.