Chcę zaoszczędzić więcej.

Chcę is the 1st person singular present tense of chcieć — I want.

The ending -ę is very common in Polish for I forms of many verbs:

• robię — I do / I am doing
• widzę — I see
• chcę — I want

The letter ę is a nasal vowel, but in everyday pronunciation its exact sound can vary depending on what comes after it.

After chcę (I want), Polish normally uses the infinitive, just like English uses to + verb:

• Chcę czytać. — I want to read.
• Chcę spać. — I want to sleep.
• Chcę zaoszczędzić więcej. — I want to save more.

So the structure is:

chcę + infinitive

This is mainly a question of aspect, which is very important in Polish.

• oszczędzać is imperfective: it focuses on the process, repeated action, or habit of saving
• zaoszczędzić is perfective: it focuses on achieving a result, saving up an amount, or completing the action

So:

• Chcę oszczędzać więcej.
  = I want to be saving more / save more regularly

• Chcę zaoszczędzić więcej.
  = I want to save more up / manage to save more

In this sentence, zaoszczędzić suggests a result rather than just the activity.

The prefix za- often helps form a perfective verb from an imperfective one.

Here:

• oszczędzać — to save, to be saving
• zaoszczędzić — to save up, to manage to save, to save a certain amount

You do not always have to translate the prefix directly. Very often, it is better to think of it as showing completion or achievement of a result.

Więcej means more.

It is the correct word when you want to talk about:

• a greater amount
• a greater degree
• doing something more than before

So in this sentence, więcej modifies the idea of saving:

• saving more

It can also be used with nouns:

• więcej pieniędzy — more money
• więcej czasu — more time

Because więcej is used for more in quantity/amount, while bardziej is usually more in the sense of more intensely / more adjectivally / more adverbially.

Compare:

• więcej pieniędzy — more money
• więcej oszczędzać — save more
• bardziej interesujący — more interesting
• bardziej ostrożnie — more carefully

In Chcę zaoszczędzić więcej, the idea is quantity or amount, so więcej is the natural choice.

Yes. Polish often leaves the object unstated when it is obvious from context.

So Chcę zaoszczędzić więcej is perfectly natural on its own. Usually the listener will understand something like:

• more money
• more than before
• a larger amount

If you want to be explicit, you can add a noun:

• Chcę zaoszczędzić więcej pieniędzy. — I want to save more money.

Usually, after więcej, the noun appears in the genitive.

For example:

• więcej pieniędzy — more money
• więcej czasu — more time
• więcej energii — more energy

So:

• Chcę zaoszczędzić więcej pieniędzy.

This is a very common pattern in Polish.

Polish word order is more flexible than English, but the neutral order here is:

Chcę zaoszczędzić więcej.

Other orders are possible, but they change the emphasis:

• Więcej chcę zaoszczędzić.
  This puts extra focus on more.

• Zaoszczędzić więcej chcę.
  This is possible in special contexts, but sounds marked or literary.

For normal everyday use, Chcę zaoszczędzić więcej is the best choice.

A rough pronunciation guide is:

• Chcę ≈ h-tsen or ktsen depending on speech style
• zaoszczędzić ≈ za-oh-shchEN-jich
• więcej ≈ VYEN-tsay

A few useful notes:

• ch sounds like the h in Scottish loch or German Bach
• cz sounds like ch in chop, but with the tongue curled slightly back
• dz sounds like ds / dz
• ć / ci gives a soft ch-like sound
• więcej starts with w, but Polish w is pronounced like English v

The sentence stress in Polish usually falls on the second-to-last syllable of each word:

• CHCĘ
• za-o-szczę-DZIĆ
• WIĘ-cej

Yes, and it is a very useful comparison.

• Chcę oszczędzać więcej
  focuses on the habit or ongoing activity: I want to save more regularly / be more saving-minded

• Chcę zaoszczędzić więcej
  focuses more on the result: I want to manage to save more / end up saving more

Both are correct, but they are not exactly the same in nuance.

Most often, it is used for money, but it can also be used for other things like:

• time
• effort
• energy

For example:

• zaoszczędzić czas — save time
• zaoszczędzić pieniądze — save money

So in your sentence, the omitted object is probably money, but context could allow something else.