How long after getting braces before you need to get a checkup / adjustment / tightening appointment?

It depends. It really depends on the type of braces and the treatment plan. Usually the patient is seen a couple weeks after the initial placement. But that varies widely based on what was actually done.
Depends. The age of patient, the severity of the problem, the need to more often monitor a patient all impact the timings of adjustments and supervision. For instance, i might well see a patient in 10 week intervals, but if oral hygiene is poor i want to see them every month. Todays modern material science in orthodontics routinely allow me to see patients between 8-12 week intervals. Good luck!
It varies. It depends on your treatment plan and type of appliances used. 2nd appointment could be in a couple of weeks or up to a couple of months after starting.