Learn Russian: Мне нужно отдыхать.

In Russian, to express "I need", you literally say "To me it is necessary". Мне is the dative case of Я (I). Words like нужно (necessary) require the person who needs the action to be in the dative case, rather than the nominative case.

Is нужно a regular verb like хотеть (to want)?

Not exactly. Even though it translates to "need to", нужно doesn't conjugate like a regular verb. It never changes to match the person in the present tense. Whether it is Мне нужно (I need), Ему нужно (he needs), or Нам нужно (we need), the word нужно stays exactly the same.

Why isn't отдыхать conjugated here?

Just like with modal verbs such as хотеть (to want) or мочь (can), when нужно is followed by an action, that second action must be in the infinitive (the dictionary form). You are literally saying "It is necessary to rest", so отдыхать keeps its basic -ть ending.

Can I use надо instead of нужно in this sentence?

Yes! Мне надо отдыхать means exactly the same thing. Надо is very common in everyday spoken Russian, while нужно is standard in both writing and speaking. They are completely interchangeable in this context.