Тебе пора бы записаться к стоматологу.

Because пора is used in an impersonal pattern:

кому-то пора + infinitive = it’s time for someone to do something

So the person goes in the dative case:

• мне пора = it’s time for me
• тебе пора = it’s time for you
• ему пора = it’s time for him

That is why the sentence starts with тебе, not ты.

Пора means something like it’s time, it’s about time, or you should really be doing this now.

In this sentence, тебе пора записаться к стоматологу means more than just a neutral statement about time. It usually suggests that the speaker thinks this action is already due or overdue.

So it often carries a gentle sense of:

• you need to
• you should probably
• it’s time you finally

This is one of the most important nuances in the sentence.

Without бы:

• Тебе пора записаться к стоматологу.
  = It’s time for you to make an appointment with the dentist.

With бы:

• Тебе пора бы записаться к стоматологу.
  = You really ought to make an appointment with the dentist / It would be about time for you to make an appointment.

The particle бы makes the statement sound a bit more:

• tentative
• softened
• suggestive
• mildly reproachful

It can feel like the speaker is hinting: you’ve been putting this off, and you really should do it.

In modern speech, many speakers would more naturally say:

• Тебе бы пора записаться к стоматологу.

That word order is often especially natural.

Записаться means to sign oneself up / to make an appointment / to get booked in.

Here are the key contrasts:

• записать = to write down, record, or sign someone up
• записаться = to sign up oneself / make an appointment
• записываться = imperfective form, focusing on process or repeated action

In this sentence, the speaker means a single completed action: making the appointment. That is why the perfective verb записаться is used.

So:

• Тебе пора записаться... = you should go ahead and get the appointment made

The -ся makes the verb reflexive, but in English you usually would not translate it as yourself.

Compare:

• записать кого-то к врачу = to book someone in with a doctor
• записаться к врачу = to book oneself in / make an appointment with a doctor

So in this sentence, записаться means that you are arranging the appointment for yourself.

It is reflexive in grammar, but the natural English translation is just make an appointment.

Because after записаться in the sense of making an appointment with a professional person, Russian often uses:

записаться к + dative

Examples:

• записаться к врачу = make an appointment with a doctor
• записаться к терапевту = make an appointment with a general practitioner
• записаться к стоматологу = make an appointment with a dentist

The idea is roughly to get oneself onto that specialist’s schedule.

Because the preposition к takes the dative case when it means to / toward / with in this kind of construction.

So:

• dictionary form: стоматолог
• after к: к стоматологу

This is a regular masculine noun pattern.

Similar examples:

• к врачу
• к доктору
• к учителю

So к стоматологу is simply the required dative form after к.

Yes. Стоматолог is the standard and very common word for dentist.

You may also hear:

• зубной врач = literally tooth doctor, understandable but less standard in many contexts
• дантист = dentist, but less common and can sound more old-fashioned or specialized depending on context

In everyday modern Russian, стоматолог is the safest and most neutral word.

Yes. Russian word order is flexible, and changing it often changes the emphasis more than the core meaning.

Possible variants:

• Тебе пора бы записаться к стоматологу.
• Тебе бы пора записаться к стоматологу.
• Пора бы тебе записаться к стоматологу.

All of these are understandable. The differences are mostly about emphasis and style.

Very roughly:

• Тебе пора бы... emphasizes you
• Тебе бы пора... is often a very natural conversational order
• Пора бы тебе... puts the focus first on it’s about time

A learner should mainly recognize that the basic meaning stays the same.

It is usually advice with a slight nudge.

It is not a direct command like:

• Запишись к стоматологу. = Make an appointment with the dentist.

And it is not as neutral as:

• Тебе нужно записаться к стоматологу. = You need to make an appointment with the dentist.

Because of пора бы, it can sound like:

• the speaker thinks this should already have happened
• the person has probably delayed it
• the speaker is gently pushing them

So the tone is often:

• concerned
• mildly insistent
• slightly reproachful, but not harsh

Yes, it is understandable and natural enough, but many speakers would probably prefer:

• Тебе бы пора записаться к стоматологу.

This version often sounds a bit smoother in everyday speech.

That said, Тебе пора бы записаться к стоматологу is still a perfectly reasonable sentence, and a learner should definitely understand it. The main thing to notice is the nuance of пора бы: you really ought to do this by now.