Learn Italian: Lei chiude la porta e prende la borsa. - She closes the door and takes the bag.

Both are regular verbs that end in '-ere' in their dictionary form: chiudere (to close) and prendere (to take). Because the subject is lei (she), we drop the '-ere' and add the regular third-person singular ending '-e'.

Is the 'ch' in chiude pronounced like the 'ch' in the English word 'cheese'?

No, in Italian, the combination 'ch' always makes a hard 'k' sound. So chiude is pronounced roughly like 'KYOO-deh'.

Why isn't there another lei before prende?

You don't need to repeat the subject. The first lei establishes who is doing both actions. In Italian, you can often drop the subject completely because the verb ending already shows who is doing the action.

Can borsa mean any type of bag?

It usually means a purse, handbag, or a personal shoulder bag. If you are talking about a simple plastic grocery bag, Italians normally use the word sacchetto or busta.