To read the Quran correctly, it is not the Tajweed that is obligatory, but the Tartil is an obligation!
Tajweed تجويد , means a set of rules for the correct pronunciation of the Arabic letters. However, the word tajweed doesn’t exist in the Quran. But the word tartīl تَرْتِيل [reading slowly and thinking about] does exist in the Quran (al-Furqan 25:32; al-Muzzammil 73:4). In the Quran the words qirāʾa قراءة [reading] (al-Araf 7:204; al-Isra 17:106; al-Inshiqaq 84:21; al-Qiyama 75:18; an-Nahl 16:98) and tilāwa تلاوة [reading and following it] (al-Baqara 2:252; al-Bayyina 98:2; Fatir 35:29; an-Naml 27:92) exists too (Abdurrahman Cetin, “Tajweed,” The Islamic Encyclopedia from Diyanet, XXXX, 253).
Therefore to read the Quran it is not tajweed, rules for the correct pronunciation, but tartīl, thus reading and thinking about, an obligation. Non Arab’s should read and think about translations of the Quran.