How would you like to learn two languages (Malay and Indonesian) for the price of one? Just joking, of course as all the lessons here are completely free. In fact although this course is on the Malaysian language, anyone wishing to learn the Indonesian language could just as well start here and then look at the differences at the bottom of each lesson (not too many, at least for day-to-day conversations, I can assure you). So I intend to develop this course further and point out to you the most important differences with the Indonesian language as we go along.
As this is a basic course I am assuming, of course, that your aim in studying the Malaysian or Indonesian language from here is to be able to understand and to make yourself understood when you travel to those countries. If your study is of an academic nature then of course you will have to go through a proper school and attend regular classes and do practical exercises to gain proficiency.
To those of you who are more interested in learning Indonesian rather than Malay from here you still have to go through these 64 lessons in Basic Malay as much of the vocabulary and structure are common to both languages. However to facilitate their study I have highlighted all the differences between the two languages here (click to go).
Before you start studying you might want to read the introduction first. Go here for an Introduction to the Malay Language.
Please note that this is just a basic course and is meant for beginners only. Advanced students of Malay are kindly requested to look elsewhere.
For those of you who have already finished the course and would like to revise constantly without having to go through the lessons all over again, I have put all the tables together here to make your revision easier (click on the smiley) 
Table of Contents
Lesson 1 Nama/Ganti nama (Name/Pronouns)
Lesson 2 Bahasa (Language)
Lesson 3 Ini...(This is...)
Lesson 4 Orang Inggeris (Englishman)
Lesson 5 Nombor 1-9
Lesson 6 Berapa? (How many?)
Lesson 7 Hari ini hari apa? (What day is it today?)
Lesson 8 Nombor 10-99. Berapa umur anda? (How old are you?)
Lesson 9 Pukul berapa sekarang? (What time is it now?)
Lesson 10 Bulan ini bulan apa? (What month is this month?)
Lesson 11 Apa khabar? (How are you?)
Lesson 12 Warna (Colours)
Lesson 13 Belum (Not yet)
Lesson 14 Hendak (Wish to)
Lesson 15 Akan (Will)
Lesson 16 Kalau (If)
Lesson 17 Questions with "Bila" (When)
Lesson 18 Jangan (Don't)
Lesson 19 Questions with "Apa" (What?)
Lesson 20 Maafkan saya (Excusing oneself)
Lesson 21 Questions with "Di mana" (Where?)
Lesson 22 Nombor (100 and above)
Lesson 23 Apa yang anda suka makan? (What do you like to eat?)
Lesson 24 Apa yang anda hendak minum? (What do you wish to drink?)
Lesson 25 Mengapa (Why)
Lesson 26 Questions beginning with "Boleh"
Lesson 27 Barangkali (Perhaps)
Lesson 28 Selalu (Always)
Lesson 29 Questions with "Siapa" (Who?)/Family members
Lesson 30 Sejuk (Cold)
Lesson 31 The present continuous tense
Lesson 32 Suka (To like)
Lesson 33 Sangat (Very)
Lesson 34 Sila (Please extending an invitation)
Lesson 35 Sudah (The past tense)
Lesson 36 Tadi (Just now)
Lesson 37 Tidak ("Not" with adjectives)
Lesson 38 Tolong (Please asking a service)
Lesson 39 Untuk (For)
Lesson 40 Yang mana? (Which one?)
Lesson 41 Dengan (With)
Lesson 42 Saya hendak beli...(I wish to buy...)
Lesson 43 Berapa harganya? (What's the price?)
Lesson 44 Short forms (nak, tak, etc.)
Lesson 45 Prepositions (to, from some place)
Lesson 46 Prepositions (to, from, for someone)
Lesson 47 Adjectives and Synonyms
Lesson 48 Betulkah? (Is that true?)
Lesson 49 Sebutan1 (Pronunciation Part 1)
Lesson 50 Sebutan2 (Pronunciation Part 2)
Lesson 51 Bukan ("Not" with nouns)
Lesson 52 Masih (Still)
Lesson 53 Use of "pernah" (ever)
Lesson 54 Alamat (Address)
Lesson 55 Dia kata... (He said that...)
Lesson 56 Badan (Parts of the body)
Lesson 57 Sakit (Sick)
Lesson 58 Making Comparisons
Lesson 59 Pekerjaan (Occupations)
Lesson 60 Penjodoh Bilangan (Classifiers)
Lesson 61 Apa ini? (What is this?)
Lesson 62 Presenting oneself
Lesson 63 Compliments and criticisms
Lesson 64 Signboard language
APPENDIX: Malay prefixes and suffixes
REVISION (Malay In A Nutshell)
Summary of the 64 lessons (for PC)
Summary of the 64 lessons (for smartphone)
From Malay to Indonesian