Learn Italian: La pentola è piena di pasta.

Because it is an adjective describing la pentola (the pot), which is a feminine singular noun. In Italian, adjectives must change their endings to match the gender and number of the noun they modify, so the base word pieno becomes piena.

Why do we use just di here instead of della?

When you are talking about what a container is full of in a general, uncountable sense, you just use the simple preposition di (of) without adding "the". You would only use della (of the) if you were referring to a specific, already-mentioned batch of pasta, like "full of the pasta we made yesterday."

Could I say piena con pasta to mean "filled with pasta"?

No, you should stick to piena di. Just like in English where we say "full of," the Italian adjective pieno almost always pairs with the preposition di. You would usually only use con (with) if you were using a past participle like riempita (filled).

Does the word pasta only mean noodles like spaghetti or penne?

In this context, it means the food we call pasta in English! However, it is a broader word in Italian that also translates to "dough" or "pastry." For example, bread dough is a type of pasta, and a breakfast pastry is often called una pasta.