Learn Croatian: Naša burza radi rano ujutro. - Our stock market works early in the morning.

Tržište is the broad term for an economic market or a conceptual marketplace. Burza refers specifically to an organized financial exchange, like the stock market or a commodities exchange.

Does radi just mean "works", or does it also mean "is open"?

Both! While raditi literally means "to work", it is the most natural and common way in Croatian to say that a business, shop, or institution is open for business. You could use je otvorena (is open), but radi is the standard phrasing for operating hours.

Could I use jutros instead of ujutro in this sentence?

Only if you wanted to change the meaning to "this morning". Ujutro is an adverb used for general or habitual mornings (meaning "in the morning"), which fits this statement about the market's standard operating hours. Jutros specifically means the morning of the current day.

Are there other common types of burza I should know?

Yes, a highly useful phrase is burza rada, which literally means "exchange of work." This is the standard everyday term for the employment office where citizens register when they are looking for a job.