Если бессонница не пройдёт, я снова запишусь к врачу.

That is a very common difference between English and Russian.

In Russian, when you are talking about a real future condition, the если clause often uses the future tense too:

• Если бессонница не пройдёт... = if the insomnia doesn’t go away
• Если он придёт... = if he comes

English usually uses the present tense in this kind of if clause, but Russian does not have to do that. So Если бессонница не пройдёт is completely normal Russian.

Because пройти is perfective here, пройдёт is a simple future form.

Yes, пройти literally can mean things like to pass, to go by, or to go through. But with illnesses, symptoms, pain, or unpleasant states, it often means:

• to go away
• to pass
• to clear up

So:

• боль прошла = the pain went away
• простуда не проходит = the cold is not going away
• бессонница не пройдёт = the insomnia will not go away

This is a very common Russian way to talk about health problems.

This is mainly about aspect.

• пройти / пройдёт = perfective
• проходить / проходит = imperfective

Here the speaker is talking about one future result: whether the insomnia will go away or not. That makes the perfective form natural:

• Если бессонница не пройдёт... = if the insomnia does not go away

If you said не проходит, that would sound more like:

• it isn’t going away right now
• it generally doesn’t go away
• it keeps not going away

So не пройдёт fits a single future outcome much better.

Записаться к врачу means to make an appointment with a doctor.

It is built from the verb записаться, which often means to sign oneself up, to register, or to book oneself in.

So:

• Я запишусь к врачу = I’ll make an appointment with the doctor

This is a standard expression in Russian. A learner should remember it as a phrase:

• записаться к врачу
• записаться на приём

Because записаться means something like to get oneself signed up / registered / booked.

Compare:

• записать = to write down, record, or sign someone up
• записаться = to sign oneself up / make an appointment

So:

• Я запишу вас к врачу = I will book you in with the doctor
• Я запишусь к врачу = I will book myself in / make an appointment

The -ся / -сь ending often gives this oneself idea.

After к, Russian normally uses the dative case when it means to, toward, or to see someone.

So:

• к врачу = to the doctor
• к маме = to mom
• к другу = to a friend

Here:

• врач is the dictionary form
• врачу is the dative singular form

So записаться к врачу literally means something like to sign up to the doctor, but idiomatically it means to make an appointment with the doctor.

Yes. Russian often omits subject pronouns when the verb ending already makes the subject clear.

So these are both possible:

• Если бессонница не пройдёт, я снова запишусь к врачу.
• Если бессонница не пройдёт, снова запишусь к врачу.

Because запишусь clearly means I will sign up / I will make an appointment, the pronoun is not strictly necessary.

Including я can make the sentence a little clearer, a little more explicit, or a little more contrastive.

Снова means again.

So the speaker is saying they will make another appointment, not the first one.

In Russian, word order is fairly flexible, so снова can sometimes move:

• Я снова запишусь к врачу.
• Я запишусь снова к врачу.
• Снова запишусь к врачу.

But Я снова запишусь к врачу is the most natural neutral order here.

Putting снова before the verb is very common.

Because Если бессонница не пройдёт is a subordinate clause, and it is separated from the main clause by a comma.

Structure:

• Если бессонница не пройдёт = subordinate clause
• я снова запишусь к врачу = main clause

Russian normally uses a comma between these clauses:

• Если X, Y.
• Y, если X.

So the comma is required.

Key stresses are:

• бессОнница
• не пройдЁт
• запишУсь
• к врачУ

A few useful notes:

• ё is always stressed
• many Russian texts write е instead of ё, so you may also see пройдет
• even if written е, the pronunciation is still ё if that word is really пройдёт

So for a learner, it is helpful to remember the spoken forms with stress:

• бессОнница
• пройдЁт
• запишУсь
• врачУ