Learn Italian: Ora abbiamo bisogno di un medico.

In Italian, the concept of needing is expressed with the phrase avere bisogno di, which literally means 'to have need of'. To say 'we need', you just use the 'we' form of avere (to have), which is abbiamo.

Why is the word di included here?

The fixed expression is avere bisogno di (to have need of). Whenever you mention exactly what you need right after the phrase, you must connect it with the word di.

Could I say dottore instead of medico?

Yes, you can. In everyday Italian, un dottore and un medico are used interchangeably to mean a medical doctor. Just keep in mind that dottore is also a general title for anyone with a university degree in Italy, while medico always refers strictly to a medical doctor.

Why is the word for 'we' (noi) missing from the beginning of the sentence?

Italian usually drops subject words like noi (we) because the verb itself tells you who is acting. The form abbiamo can only mean 'we have', so adding noi is unnecessary unless you really want to emphasize who has the need.