Learn Polish: Nasz nowy dom jest większy.

The basic word for "big" is duży. This is an irregular comparative, meaning the root changes completely—just like in English where "good" becomes "better". For "big", the root changes from duż- to większ-, and then you add the standard adjective ending.

Why does większy end in -y?

Comparative adjectives still act like regular adjectives, meaning they have to match the noun they describe. Because dom (house) is a masculine noun, większy takes the masculine singular ending -y. If we were talking about a feminine noun, like sypialnia (bedroom), it would be większa.

I always mix up the forms of "our". Why is it nasz here instead of nasze?

Possessive words like "my", "your", and "our" must match the gender of the object being owned, not the gender or number of the owners. Even though "we" (the owners) are plural, the word dom is singular and masculine, so we use the masculine singular form nasz.

Do I need to add a word for "than" after większy?

Not if the context is already clear! Just like in English, you can simply end the sentence at "bigger". If you wanted to add what it is bigger than, you would typically use the word niż (than) or od (from) followed by the other object.