It is a bit odd, but it's a standard phrase at Starbucks (and some other stores).
"To get started" is "to start doing" or "to start making." I might say "I'm going to get started on my homework." or "I have to go home and get started on dinner." or "You should get started on cleaning the kitchen." Which means I'm going to begin my homework, or that I'm going to begin cooking dinner, or you should start doing the dishes.
At Starbucks, "What should we get started for you today?" means "What do you want to order?" but it also has the implication that they are going to start making your drink or food, but it might take some time for them to finish your order and you may need to wait a few minutes. ((It also gives you the feeling that your order needs special care - it can't be rushed through, it is something that needs to be started and then completed, because you are important and special.))