Jestem uczniem i muszę codziennie uczyć się nowych słówek.

Because Polish usually does not need subject pronouns when the verb ending already shows the person.

• jestem = I am
• muszę = I must / I have to

So Ja jestem uczniem... is possible, but it sounds more emphatic, for example if you are contrasting yourself with someone else.

Because after jestem and other forms of być used to describe someone’s role, identity, or status, Polish normally uses the instrumental case.

So:

• uczeń = nominative, the basic dictionary form
• uczniem = instrumental singular

That is why you say:

• Jestem uczniem = I am a student / pupil

not normally Jestem uczeń.

Uczniem is the instrumental singular of uczeń.

This is worth learning as a case pair:

• uczeń → uczniem

The change is not fully predictable from English, so it is best to memorize it together with the base noun.

Muszę is the first-person singular form of musieć, meaning to have to / must.

So:

• muszę = I must / I have to

After muszę, Polish uses an infinitive:

• muszę uczyć się = I have to learn / study

This works much like English must + verb or have to + verb.

Because uczyć się is the verb meaning to learn / to study.

The się is an essential part of this verb. Without it, uczyć usually means to teach.

Compare:

• uczyć kogoś = to teach someone
• uczyć się = to learn / to study

So in this sentence, się is not optional.

In dictionaries and basic learning, it is very common to learn this verb as a unit:

• uczyć się

So in a sentence with an infinitive, placing się after the infinitive is very natural.

You should think of uczyć się as one expression. Polish word order is somewhat flexible, but for learners the safest choice is to keep them together.

Because uczyć się usually takes the genitive case.

A useful question pattern is:

• uczyć się czego? = to learn/study what?

So:

• nowych = genitive plural of nowe
• słówek = genitive plural of słówko

Together, nowych słówek is genitive plural.

Słówko is a diminutive form of słowo.

• słowo = word
• słówko = literally something like little word, but very often it just means word / vocabulary item in a friendly, everyday way

In language-learning contexts, słówka is extremely common for vocabulary words.

So nowych słówek sounds very natural here. You could also hear nowych słów, which is more neutral.

Because uczyć się is imperfective, and it fits repeated or ongoing activity.

Here you have codziennie = every day, so the idea is habitual:

• muszę codziennie uczyć się = I have to learn/study every day

By contrast, nauczyć się is usually perfective, meaning to learn/master something completely, often with a sense of completion.

So:

• uczyć się = ongoing, repeated learning
• nauczyć się = learn/master something successfully

With codziennie, the imperfective verb is the natural choice.

The word order is not completely fixed. Polish is more flexible than English.

These are all natural:

• Muszę codziennie uczyć się nowych słówek.
• Codziennie muszę uczyć się nowych słówek.
• Muszę uczyć się nowych słówek codziennie.

The version in your sentence is a neutral, very common order.

Yes, i is the normal coordinating conjunction meaning and.

In this sentence, it simply joins two facts:

• Jestem uczniem
• muszę codziennie uczyć się nowych słówek

Using a would change the feeling. A often suggests contrast or a switch of topic, something like and/but/as for depending on context. Here i is the most straightforward choice.

A few helpful pronunciation notes:

• cz sounds like ch in chop
• sz sounds like sh in ship
• ń sounds like ny in canyon
• się has a soft ś sound, something like sh but softer
• final ę is often pronounced less fully in everyday speech

Rough English-style approximations:

• Jestem ≈ YES-tem
• uczniem ≈ OOCH-nyem
• muszę ≈ MOO-she
• uczyć się ≈ OO-chich shye
• słówek ≈ SWOO-vek

These are only approximations, but they can help you get close.