Die Verben «мочь» und «хотеть»

"Can" and "want" are the two verbs you will hook most other verbs onto. In Russian both are followed by a plain infinitive, with no "to" in between: Я могу́ прийти́ - "I can come", Я хочу́ спать - "I want to sleep".

The г that turns into ж

In мочь the stem consonant alternates: г in "I" and "they" (могу́, мо́гут), ж everywhere in between (мо́жешь, мо́жет, мо́жем, мо́жете). The past tense is short and a little odd: мог, могла́, могли́.

хотеть changes pattern halfway

The singular has ч and first-conjugation endings (хочу́, хо́чешь, хо́чет); the plural has т and second-conjugation endings (хоти́м, хоти́те, хотя́т).

English "can" is two Russian verbs

Мочь is about circumstances or permission: Я не могу́ сего́дня, я рабо́таю - "I can't today, I'm working". A learned skill is уме́ть: Я уме́ю пла́вать - "I can swim (I know how)". So "I can't swim today, I have a cold" is Я не могу́ пла́вать, while "I can't swim at all" is Я не уме́ю пла́вать. For languages Russians usually just say Я говорю́ по-ру́сски rather than "can speak".

Wanting politely

Я хочу́ ко́фе is grammatical, but said to a waiter it sounds blunt. In a café people say Мо́жно ко́фе, пожа́луйста? or Я бы хоте́л(а) ко́фе. With friends, хоте́ть is fine and often drops the motion verb altogether: Хочу́ в Со́чи! - "I want to go to Sochi!"