By learning these rules, you should improve your conversational skills significantly without putting in too much effort.

I'm not a native Indonesian speaker so if you think I wrote something wrong, please don't hesitate to leave a comment.

Placing words in the right order in Indonesian
Most of the time, you can place words in the same order as you would in English: Subject + Verbs + Object.

There are a few exceptions though:

Personal pronouns:
They are placed after the noun. For instance: "Your cat" = "Kucing kamu" (literally "Cat you")

Adjectives:
They are also placed after the noun. For instance: "New car" = "Mobil baru" (literally "Car new").

Talking about a future event in Indonesian

→ Lagi is the most used in informal conversations. Placed before the verb, it generally describes a short action (eating, watching a movie, going shopping, etc). For instance, "I'm eating" = "Aku lagi makan"

→ Sedang is more formal and it can be used for both short and long actions (learning a language, studying, working in a company). It is also placed before the verb. For instance, "Saya sedang cari kerja" = "I'm searching for a job".

Giving a date and the time in Indonesian
How to express the time or a date is different in English and in Indonesian. In the latter, before the time marker, you must add a noun to specify what the marker is about. For instance:

Time:
Jam + numerical. Example: "Jam 3" = "3 o'clock"

Day:
Hari + day. Example: "Friday" = "Hari Jumat"

Months:
Bulan + month. Example: "November" = "Bulan November"

Year:
Tahun + year. Example: "2017" = "Tahun 2017"

Specific date:
Tanggal + date. Example: "January 10th" = "Tanggal 10 Januari"

Using prefixes and suffixes in Indonesian

Those suffixes and prefixes modify the meaning to the root word according to specific rules. If you learn these, you will be able to improve your vocabulary significantly.

Based on my experience, this is the hardest part of learning Indonesian grammar. As a beginner though, you only need to understand the concept and to be able to recognize when there is a prefix or a suffix.

These are the most common prefixes:

Ber-: Usually, it is added to a noun, a verb or an adjective to form a verb whose meaning is "Having the attributes of...".
Example: Berkacamata = To wear glasses (from "kacamata" = "glasses").

Men-: Most often, it is added to a noun, a verb or an adjective to form a verb whose subject is actively doing something.
Ex: Mengajar = To teach (from "ajar" = "teaching").

Nge-: A very informal way to create verbs from nouns or adjectives. It is popular among young people to adapt English words to Indonesian.
Ex: Ngebeer = to drink beer, Ngechat = to chat, Ngecharge = to charge a phone, Ngegym = to go to the gym, etc.

Pen-: Generally, it is added to a noun, an adjective or a verb to form the noun of a person doing/or being as the root word.
Ex: Pemalu = A shy person (from "malu" = "shy").

Ter-: The easiest one, it is a superlative that can be translated by "the most":
- "Terkenal" = "The most famous" (literally, "the most known")

Di-: Adding "Di" before a verb is how to form the passive voice in Indonesian.
- Dipegang = to be grabbed (from pegang = to grab)

These are the most common suffixes:

-nya: A very common and useful suffix that you can add to a noun. It means "its", "his", her" or "the".

This is not completely true. Doubling a word does not always work. It is adequate to translate an expression like "a bunch of", but it is better to avoid in any other cases. In most informal speeches, the best thing to do is to leave the word as it is. Your interlocutor should understand if there is a plural from context alone. To help with that, you can use numbers or adverbs like "banyak" (a lot) or "beberapa" (several).

Asking questions
The easiest way to ask yes-no (or closed) questions in Indonesian is to add a question mark at the end of a sentence. For instance: "Is he cooking?" = "Dia lagi masak?" (literally "He is cooking?").

You can also start your sentence with the formal "Apakah", or its shortened version "apa". For instance, "Is he your older brother?" = "Apa (kah) dia kakak kamu?". "Apakah" means "Is..." or "Are...".

When using an interrogative pronoun such as "Apa" ("What"), "Berapa" ("How many"/"How much"), "Bagaimana" ("How"), "Siapa" ("Who"), etc, you can place it either at the end or at the beginning of your sentence.

A major difference with English is that the word "Apa" (What) cannot be used with people's name or numbers. To translate "What is your name?", you cannot say "Apa nama kamu?" (literally: What name you?) but "Siapa nama kamu ?" (literally, Who Name You). Similarly, to translate "How old are you?", you cannot say "Apa umur kamu?" (literally, "What age you?") but "Berap a umur kamu?" (literally: "How much age you?").

Note that the interrogative pronoun "apa" and the shortened version of apakah, "apa", are two different words with different meanings. 

→ In formal speeches, you will use "Bukan" in front of a noun or a pronoun, and "Tidak" in front of an adjective or a verb.

For instance:
Kamu bukan ayah aku = You are not my dad
Aku tidak mau keluar = I don't want to go out

→ When speaking informally to friends, you will often replace "Tidak" with "Ngak", "Gak" or "Tak".

For instance:
Aku gak mau keluar = I don't want to go out

→ When giving a negative order, you should use "Jangan" at the beginning of the sentence.

For instance:
Jangan bilang itu! = Don't say that!

→ When talking about your experiences, your wishes or your tastes, it is better to avoid using a strict negation. The words "Belum" ("Not yet") and "Kurang" (Less) can replace "Tidak" or "Bukan" to form more polite sentences.

For instance:
Instead of "Aku tidak suka makanan Jepang" ("I don't like Japanese food"), say: "Aku kurang suka makanan Jepang" (literally, "I less like Indonesian food").

Instead of "Aku tidak mau nikah" ("I don't want to marry"), say: "Aku belum mau nikah" ("I don't want to marry yet").

→ Generally speaking, "to be" is not needed (unless you want to sound formal). This is particularly true when it is used to describe something/someone or to give characteristics.

For instance:
I am a man = Aku cowok (literally, "I man").
He is fat = Dia gemuk (literally, "He fat").

→ You can use "adalah" to make a statement that two things are equivalent.

For instance:
"Rusia adalah negara terbesar di dunia" = "Russia is the largest country on earth"

→ Another use of "Adalah" is to say that something or someone is part of a group/structure.

For instance:
"Dia adalah pengawai pemerintah" = "He is a government employee".

Indicating presence
To say "There is" or "There are", you should use the popular word "Ada" (also: "Tidak ada" = "There is not"). It is placed at the beginning of sentence such as:
"Tidak ada rokok lagi di toko" = "There are no more cigarettes at the store"
"Ada siapa di rumah kamu?" = "Who is at your house?" (literally, "There is who at your house?")

In informal conversations, "Ada" can also mean "to be present". For instance: "Dia ada di rumah" = "He is at home".