Understanding "英語のレッスンしないとな"

Question

The sentence

英語のレッスンしないとな。

apparently translates to

I need English lessons.

However it seems to me this sentence means something like

Don't do English lessons, ???

so I'm pretty terribly confused.

Other Questions:

1. What is the とな at the end signify? My best guess it something like "it's been said, eh?" (with と being used as the quotation particle).
2. If I'm right on (1), does this sentence mean something like "It's been said I don't do English lessons, eh?"

Answer 1

まずは少し文法を勉強しないとな。
Which is to say: to understand this construction, you "need to" learn how to express “need to” in Japanese. I recommend you read the following comprehensive Guide To Japanese article from start to finish: Expressing “must” or “have to”.

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As you'll see, there are many forms of what is essentially the same construction. The first half of that article covers the unabbreviated forms, which includes the と-form with 2 examples:

毎日学校に行かないとだめです。
Must go to school everyday.
宿題をしないといけない
Have to do homework.

The second part of that article mentions their abbreviated forms, which are actually a lot more commonly used than their verbose counterparts—in spoken/casual language at least. Here you'll find the abbreviated と-form too:

The 「と」 conditional is also used by itself to imply 「だめ／いけない／ならない」.

学校に行かないと。
Gotta go to school.