I dont think there is way to achieve what you are asking for, and there is a good reason for it too. You do not want the start and finish dates to always be matching your baseline dates. If we were to do that then all the future tasks on your projects will always be on schedule, even though that may not be the truth. You want the project to be ‘fluid’ so you can identify future delays in the project early on and do something about it if possible. Nothing makes management more frustrated then knowing on the Friday before, that the task scheduled to start on Monday is now delayed, since they cannot really do anything about it now.
However, for the actual start/finish date there is an option. If you go to File -> Options -> advanced and then scroll all the way down, you will see an option – Edits to the total task% complete will be spread to the status date. If you check this then when you mark a task 100% complete it will mark the actual finish date of the task to the ‘status date’.