Since you'll be getting to know someone, you will still want to use the slightly formal forms that end in ます. Here are some examples: 趣味は何ですか？ (What are your hobbies) 今日はどう過ごしていましたか？ (How did you spend the day today) 彼女・彼氏はいますか？　(Do you have a boyfriend/girlfriend; This can be personal, but it's actually a pretty common question I have been asked, even by strangers) 何が好き・嫌いですか？　(What do you like/dislike) 海外へ行ったことありますか？　(Have you ever been abroad) ＊～たことありますか？ (this construction means "have you ever done~" and is useful for creating all kinds of different questions.) 出身はどこですか？　(Where are you from) 何才ですか？　(How old are you) These are some of the major ones that I can think of for the moment. If I think of more, I will let you know. I've used all of these examples before and found them very useful, however I can't speak with complete authority since I am not a native speaker of Japanese.

Thanks, Sheltie! I'll add these to my list :)

Here are some examples. （１）どんな食べ物が好きですか？　What kind of food do you like? （２）休みの日は何をしていますか？　How do you spend in your days off? （３）アメリカへ行ったことがありますか？　or　アメリカにきたことがありますか？　Have you been to America? （４）どこに住んでいますか？　Where do you live? I wrote them in only hiragana. （１）どんなたべものがすきですか？ （２）やすみのひはなにをしていますか？ （３）アメリカへいったことがありますか？ （４）どこにすんでいますか？