Aku bikin daftar belanja di kertas kecil biar nggak lupa.

Aku is the casual, personal I used with friends, family, people your age, or in relaxed situations. Saya is more formal/polite and common in workplaces, with strangers, or in official contexts.
Both can work grammatically here, but the tone changes:

• Aku bikin daftar belanja... = casual, everyday speech
• Saya membuat daftar belanja... = more formal/neutral

Bikin means to make / to create / to do. It’s very common in spoken Indonesian and informal writing. The more formal equivalent is membuat (or sometimes menyusun for “compile/arrange a list”).
So:

• Aku bikin daftar belanja = I made a shopping list (casual)
• Aku membuat daftar belanja = I made a shopping list (more formal)

daftar belanja is the standard collocation for shopping list.

• belanja = shopping (activity)
• belanjaan = shopping items / the things you bought
  So daftar belanjaan would lean toward “list of items (to buy/bought)” and may sound less standard than daftar belanja.

di is a general preposition for location: in/at/on, depending on context. With paper, English often prefers on, but Indonesian commonly uses di:

• di kertas = on a piece of paper (natural Indonesian) You could also say di selembar kertas kecil (“on a small sheet of paper”) for extra specificity.

Indonesian often omits classifiers/measure words when they’re not needed. di kertas kecil is fine and natural.
If you want to be more specific:

• di selembar kertas kecil = on a small sheet of paper
• di kertas kecil itu = on that small piece of paper

biar means so that / in order that / so (I) don’t… and is very common in casual speech.
supaya and agar are slightly more formal/neutral.
In this sentence:

• biar nggak lupa = so I don’t forget (casual)
• supaya/agar tidak lupa = so that I don’t forget (more formal)

nggak (also spelled gak) is informal not. tidak is the standard/formal negative.
Both are correct, but the register changes:

• biar nggak lupa = casual
• agar tidak lupa = more formal

lupa means to forget. It can be used:

• intransitively: Aku lupa. = I forgot.
• with an object: Aku lupa namanya. = I forgot his/her name.
  Here it’s intransitive/implicit: nggak lupa (belanjaannya/apa yang harus dibeli) = not forget (what to buy).

Both are possible in English, but Indonesian bikin daftar focuses on creating/making the list, not the physical act of writing. If you want to explicitly say “wrote,” you could say:

• Aku nulis/menulis daftar belanja... = I wrote a shopping list...

You can reorder somewhat, but some orders sound more natural than others. Common alternatives:

• Biar nggak lupa, aku bikin daftar belanja di kertas kecil. (Reason first)
• Aku bikin daftar belanja biar nggak lupa, di kertas kecil. (Less clean, but possible) The original order is very natural: main action → location → purpose.