You are a very responsible and respectful person for inquiring about whether wearing your current clothes would be disrespectful or not to the synagogue. I really have to hand it to you, not everyone is that respectful.
It depends on what day of the week you plan on visiting to the synagogue. If you're planning on visiting during a weekday, then your normal clothes, provided that they aren't indecent, will suffice. There are plenty of Orthodox Jews who, during the week, wear their normal working clothes to the services. If you're planning on going on a Shabbat, however, your white shirt and black slacks should suffice, since you said you don't have anything else. That's what I wear everyday (black and white), and that's what I wear whenever I go to synagogue, for Shabbat even. (I live in a place that does not have an Orthodox synagogue, so I do all my praying at home, but when I do get the chance to attend synagogue in another city I do so.)
(It's a pretty widespread myth that Orthodox Jews wear suits everyday to synagogue.)
As for not owning a kippah, most synagogues will have a bin by the door with kippot for visitors to use. So you could use on of those. Otherwise, a different head covering would be good. Your beanie might not be such a bad option, but during a weekday it wouldn't be strange to even wear a ballcap!
Remember to talk to the rabbi of the synagogue you're planning to visit and explain why you want to visit before you show up to the synagogue. This way the rabbi will know your motives and will be able to help you, and he won't be surprised and accidentally stand-off-ish when you show up. Make sure to explain to him that you are not a Jew (that you only have a Jewish father) but that you're interested in converting to Judaism and want to see what it's like "inside".
During the services, you might explain to your neighbors in synagogue that you're interested in learning about Judaism and more importantly, that you're not Jewish. This way they can help you if you have questions about the service, and any polite person should be happy to teach you.
If you plan to go on Shabbat, remember that since you're not Jewish according to Jewish Law, then there should be no problem with you driving to synagogue during the Shabbat, and no one would judge you for doing so. It might be a good idea, though, to park a little ways off and walk up to the synagogue, and not park in the parking lot.
I would actually recommend visiting on a Shabbat, so that you will get a full synagogue experience. Additionally, the prayers tend to be recited slower, since no one has to go to work right after services. Also, on Shabbat the Torah is read, whereas during the week the Torah is only read on Monday and Thursday.
But as mentioned above, the most important thing is that you speak with the rabbi of the synagogue you want to attend.
Good luck in your journey and welcome to Mi Yodeya. If you have any other questions, feel free to ask me in the comments of this answer.
For further information, see this question and answer elsewhere on this site.
--- Added by DanF ---
If no one knows you there, you actually should make it known that you're a Gentile! This may sound strange, but once, in my shul, we had 10 men which included a stranger. We assumed that he was Jewish and we had a minyan. It was only when we reached Torah reading, someone asked him if he wanted to be called to the Torah. That's when he said, "I'm not Jewish". So, we had actually performed parts of the service assuming we had a minyan, when we really didn't. So, you - or rather the shul - wants to avoid this situation!
