Beginner question about kanji
 

What can I do if I stumble upon a kanji that I just don't know what it means? And of course I don't know the hiragana for it either.

You look it up in the dictionary. There are a number of kanji lookup methods. WWWJDIC: Kanji Selection

The "selection type" drop-down lists a number of lookup methods.

Now, if it's on the computer, the easiest way is, within Firefox or Chrome, use the add-on Rikai-chan or Rikai-kun, respectively, and just mouseover the kanji.

I often use my Kanji Learner's Dictionary with the SKIP method, but I also use my electronic dictionary to look some up, too, first. There are, as I said, many different methods.

Now, if you mean without using a reference, you can't really know, but sometimes you can guess at its meaning or pronunciation. Meaning through context. Pronunciation sometimes through parts of the kanji. Oftentimes, parts of a kanji hint at its pronunciation. You will discover this over time, which will help in learning more kanji. I noticed this really started paying off once I knew about a thousand kanji already. Learning the reading of the next thousand was much easier because of this.