おぼえる is an ichidan verb, therefore its polite form is おぼえます, not おぼえります. As a general rule, verbs that end with -eru or -iru are ichidan (though there are a few exceptions, e.g. かえる, はいる).
わたしは can be omitted. In Japanese, the subject is often dropped because it's clear from context, so using わたしは all the time sounds overly verbose. If you make a statement without a subject, it's typically assumed that you're talking about yourself, while if you ask a question without a subject, it's assumed to be the person you're talking to. So you can shorten your answers to simply いいえ、まだそうじしていません or simply まだしていません.
For はなす, the particle と is more suited for indicating the person you're talking "with" (せんせいとはなしました). に can be used when you're asking someone a question (せんせいにききます).
If someone asks a question with もう (e.g. "have you already done your homework"), you can answer in the affirmative (yes, I have *already* done it: *もう*しました) or in the negative (I haven't done it *yet*: *まだ*していません). Just like in English, the words in stars are different depending on whether you answer yes or no.