Learn Hardcore Japanese: hayai densya ha benri desu ga, yasuku ha arimasen. - Fast trains are convenient, but not cheap.

速い電車 is an attributive phrase where the i-adjective 速い (“fast”) directly modifies the noun 電車 (“train”), giving “fast train.”
In Japanese, when an i-adjective modifies a noun, you use its plain form (here 速い) without adding です, because です only appears in predicative (ending) positions, not before a noun.

Why is 電車 marked with は instead of が here?

は marks the topic of the sentence (“As for fast trains…”), setting the frame for the comment that follows.
が would mark the subject in a neutral way, but using は signals you’re talking about trains in general or contrasting them with something else.

Why does the first clause use 便利です? Isn’t 便利 an adjective like 速い?

便利 is a na-adjective, not an i-adjective. Na-adjectives require the copula です/だ when used predicatively (“is convenient”).
In polite speech you use です, so 便利です means “is convenient.”

What role does the が in the middle play? It doesn’t look like the subject-marker が.

This が is a conjunction particle meaning “but” (similar to けれども, しかし).
It links two clauses: “Fast trains are convenient, but…”

Why is it 安くはありません instead of simply 安くありません?

The extra は after 安く is the contrastive topic marker, highlighting “as for being cheap, that’s not true.”
安くありません would just state “is not cheap,” but 安くはありません adds a nuance of contrast or emphasis (“it’s not cheap (though it’s fast)”).

How do you form the negative of an i-adjective like 安い?

Drop the final い, then add くない in plain style or くありません in polite style.
- Plain negative: 安い → 安くない (“is not cheap”)
- Polite negative: 安い → 安くありません (“is not cheap”)

Is 電車 singular or plural here? How do you tell?

Japanese nouns don’t change form for singular/plural. 電車 can mean “(a) train,” “trains,” or “the train” depending on context.
In this sentence it’s a generic statement about trains in general (“Fast trains are convenient but not cheap”).