There are no standard sizes or stocks. A "flier" can be any size on any stock.
You can use whatever size and stock you want. The word "flier" on its own designates the intended usage, not the appearance in any way.
To relate this to your profession.... this question is similar to asking, "What ethnicity is a patient?" There is no "standard" answer.
Use whatever size you feel works best and whatever stock you feel conveys the message you wish to convey. For a medical practice you probably don't want to use oversized fluorescent green stock. It's that sort of aesthetic determination that would indicate a stock which works for you.
If you are uncertain about this, whoever is printing the flier may be able to assist with these choices. If you are doing this yourself, look for fliers you like and try to mimic the qualities you find appealing, or hire someone to determine this for you based upon your specific needs.