Learn Swedish: Vi har inte råd med en ny bil.

You are right that ett råd commonly means 'a piece of advice'. However, the phrase att ha råd is a fixed expression meaning 'to be able to afford'. In Swedish, råd also historically refers to having the means or resources to do something.

Why do we need the preposition med? In English we just say 'afford a new car'.

When the phrase ha råd is followed by a noun (like en ny bil), you must always use med to connect them. You can think of it literally as 'having the means with' an object.

How would I change this sentence to say 'We cannot afford TO BUY a new car'?

If you want to follow the phrase with a verb instead of a noun, you drop med and use att plus the infinitive verb. It becomes: Vi har inte råd att köpa en ny bil.

Why isn't there an article before råd?

Because ha råd is a set phrase, it doesn't take an article. It works just like other common expressions that pair ha with a bare noun, such as ha tid (to have time) or ha lust (to feel like doing something).