Learn Italian: Mi piace disegnare e dipingere.

In Italian, you always use the singular form piace when talking about actions (verbs), even if you list more than one. You only switch to the plural piacciono if you are talking about plural objects. Because disegnare and dipingere are actions, piace stays singular.

Why doesn't the sentence start with Io (I)?

The verb piacere works backwards compared to English. It literally means 'to be pleasing'. So instead of saying 'I like to draw', Italians literally say 'To draw is pleasing to me.' The word Mi means 'to me', which is why we do not use Io.

Why don't we need a word for 'to' before disegnare and dipingere?

The verbs disegnare and dipingere are in their basic dictionary form (called the infinitive). This form already has 'to' built into its meaning. Disegnare means 'to draw' all by itself, so you can place it directly after piace without adding any extra words.

Why do the two action words have different endings (-are vs -ere)?