Learn Russian: Новый учитель всегда нам помогает.

In English, "us" is a direct object ("helps us"). But in Russian, the verb помогать (to help) always requires the Dative case. It helps to think of it as giving help to someone. Because of this rule, the pronoun мы (we) changes into its Dative form, нам.

Is it okay to say помогает нам instead of нам помогает?

Yes! Russian word order is flexible. Новый учитель всегда помогает нам is perfectly correct. Placing the pronoun нам right before the verb is just a very common and natural-sounding way to phrase it in everyday speech.

Since учитель ends in a soft sign (-ь), how do I know it takes the masculine adjective новый?

Nouns ending in a soft sign (-ь) can be either masculine or feminine, and you often just have to memorize their gender. However, учитель refers to a male teacher (or the teaching profession in general), so it is a masculine noun. Therefore, it takes the standard masculine adjective ending -ый.

What is the dictionary form (infinitive) of помогает?