Hi there!
Given your English translation, I would rephrase the first sentence as:
私はレストランで食べたが、後で食べるためにもっと食事を注文しました。
I think this is a more natural way to express that thought. Also, you wrote 食べります, and since the verb 食べる is an Ichidan Verb, the 「り 」is not necessary.
The second sentence has an extra sentence nominaliser (こと and の) but we only need one. In this case, we use の when using 上手 or 下手, so it ends up being:
私は日本語を話すのが上手になりたいから、今日も勉強しなきゃ。
I added も after 今日 in order to make it sound more natural.
As for sentence number 3, you missed the small っ in the verb 入る, which is a Godan verb. Since the person above this post is asking, I’ll put a link of a thread I made about Japanese Conjugations in the reply to that post. I think you should check it out too .
Anyway, here’s the corrected 3rd sentence:
この場所に入ってはいけません。
HTH!