Не успела я включить первую передачу, как машина остановилась, и инструкторша сказала снова нажать на педаль газа.

Both mean the same basic thing: I didn’t manage / barely had time (to do X).
Putting the verb first (Не успела я…) is a common narrative style in Russian; it sounds more literary/dramatic and emphasizes the “hardly had I…” feeling. Я не успела… is more neutral and conversational.

It’s a set construction meaning Hardly had … when … / No sooner … than ….
Structure:

• Не успел(а/о/и) + subject + infinitive,
• then как + past event.

So: Hardly had I engaged first gear, when the car stopped…

Because the subject is я (I), and past tense verbs in Russian agree in gender/number:

• я (female speaker) → успела
• я (male speaker) → успел
• мы → успели

The sentence implies the speaker is female (or it’s written from a female narrator’s perspective).

It usually means to manage to do something in time / to have time to do it. With не it often means not to have time. In this construction it’s closer to barely had time:

• успеть сделать = manage to do (before it’s too late)
• не успеть сделать = not manage / not have time
• не успела сделать, как… = hardly had done / barely started doing, when…

включить (perfective) treats it as a single completed action: to engage/turn on (once). That fits the “hardly had I done it” sequence.

включать (imperfective) would sound like an ongoing/repeated process and is less natural here. You might see imperfective if the meaning were “hardly had I started engaging…” but the idiom strongly prefers the perfective.

передача here means gear (as in a car’s gearbox).
первую передачу = first gear in the accusative case, because it’s the direct object of включить.

Adjective agreement:

• feminine noun передача
• accusative feminine singular → первую

Russian typically sets off this construction with a comma:
Не успела …, как …
The comma marks the boundary between the first event and the immediate follow-up event. It’s treated like a fixed “as soon as” linkage and is normally punctuated this way.

Yes. остановиться is the perfective “to stop” (intransitive), formed with -ся.

• машина остановилась = the car stopped (by itself / came to a stop)

Compare:

• остановить машину (perfective, transitive) = to stop the car (someone stops it)
• машина остановилась (intransitive) = the car stopped

Because машина is feminine, and past tense agrees with the subject:

• машина остановилась
• автомобиль остановился (masculine)
• такси остановилось (neuter)

инструкторша is a feminine noun meaning female instructor. It can sound informal/colloquial. In many contexts, people also use инструктор for a woman (profession titles are often used in masculine form as “default”), but инструкторша makes it explicitly female and sometimes feels a bit chatty or “everyday speech.”

After verbs like сказать / попросить / велеть / приказать, Russian often uses an infinitive to express an instruction, especially when it’s clear the instruction is directed at the listener:

• сказала нажать = told (me) to press

A more explicit version is possible:

• сказала, чтобы я снова нажала… = told (me) so that I would press again
  This sounds more spelled-out and can add emphasis; the infinitive version is more compact and natural for direct instructions.

Because the instruction is for a single action: press (once).
Imperfective нажимать would suggest repeated/continuous pressing:

• нажать на педаль = press the pedal (one press)
• нажимать на педаль = be pressing / press repeatedly

The verb нажать commonly uses на + accusative for what you press:

• нажать на что? → на педаль (accusative)

газа is genitive singular modifying педаль:

• педаль чего? → педаль газа = gas pedal (accelerator)

So the whole phrase is: на (what I pressed) + [gas pedal] → на педаль газа.