Мне сначала было страшно садиться за руль одной, но рядом с инструкторшей я чувствую себя спокойнее.

Мне было страшно is an impersonal construction. Instead of saying “I was scared” with a personal subject (я), Russian often says literally “To me it was scary.”

• мне = dative case (“to/for me”)
• было страшно = “it was scary” (no grammatical subject)
  This is very common with feelings/states: мне холодно, мне грустно, мне интересно, etc.

сначала means “at first / in the beginning.” It’s an adverb and can move around for emphasis, but commonly appears early:

• Мне сначала было страшно… = “At first, I was scared…”
  You could also hear Сначала мне было страшно… with a slightly more “scene-setting” feel.

Both are possible, but they feel a bit different:

• садиться (imperfective) emphasizes the process / repeated idea / getting oneself to do it: “to get (oneself) to sit down / start driving.”
• сесть (perfective) emphasizes a single completed action: “to sit down (once), to get behind the wheel (once).”
  In “It was scary to start driving,” садиться often sounds more natural because it highlights the psychological hurdle.

Literally, садиться за руль is “to sit down behind the steering wheel,” i.e. “to get behind the wheel / to start driving.”
With за, Russian uses the accusative to show movement into a position:

• за руль (accusative) = to/into the “behind-the-wheel” position
  Compare:
• сидеть за рулём (instrumental) = to be sitting/driving (already there)

Because the verb implies movement into that position: садиться = “to sit down (to take a seat).”
Rule of thumb for за:

• за + accusative = movement/change of position (садиться за стол, вставать за станок)
• за + instrumental = location/state (сидеть за столом, работать за станком)

одной is the instrumental case feminine singular of одна, and it means “alone / by myself.”
Russian often uses instrumental to express “acting as / being in the state of”:

• ехать одной = to travel alone
• жить одной = to live alone
  Because the speaker is presumably female, it’s одной. A male speaker would say одному.

Yes. The only thing that must change is gender-marked одной:

• Female: садиться за руль одной
• Male: садиться за руль одному
  Everything else can stay the same.

но = “but,” introducing contrast.
рядом с means “next to / beside” and requires the instrumental case:

• рядом с инструкторшей (instrumental)
• рядом с другом, рядом с домом, etc.

инструкторшей is instrumental of инструкторша, a feminine noun meaning “(female) instructor.”
It’s understandable and common in speech, but style-wise it can feel slightly colloquial to some speakers. Alternatives:

• с инструктором can be gender-neutral in practice (often used even if the instructor is a woman)
• с инструктором-женщиной is explicit but wordy
  In driving contexts, you’ll hear both инструктор and инструкторша.

This often shows a shift from a past initial stage to a current/general reality:

• Мне сначала было страшно… = “At first, I was scared…” (back then)
• …но рядом с инструкторшей я чувствую себя спокойнее. = “but with the instructor next to me I feel calmer.” (now / in that situation generally)
  If you want it fully in the past narrative, you can say: я чувствовала себя спокойнее.

чувствовать себя means “to feel (oneself), to feel (in some condition).”
себя is the reflexive pronoun (“myself/yourself”), used because the feeling is about the subject’s own state:

• Я чувствую себя хорошо/плохо/уверенно/спокойно.
  You generally can’t drop себя here.