I have an old Takara road bicycle that I got at Goodwill eight years ago. No model number but it was made in 1983. It ahd been altered and was a five speed when I got it. It needs new sprockets, and bike shops keep telling me that they have nothing that will fit it and they don't have the vaguest idea how I would find or recognize a part that would fit; they purport not to even know how they'd know what sprockets in their box of parts would fit it.
How do I know what parts to order?
I also need to know how to know what frames the handlebar stem and saddle stem that fit this bike would fit.
Takara's are Japanese, so they shouldn't have anything too unusual on them. I'm a little confused why a shop would say it is hard to get parts for this unless there is something unique about the bike.
You can get 5 speed or 6 speed freewheels online easily. If you have index (click) shifting, you should match the count and brand if possible. But if it is friction shifting, any 5 or 6 should work. I'd get a new chain as well if you're replacing the freewheel.
Front chainrings can be a little trickier as there is different bolt diameters. But again, I'd be surprised this bike had anything unusual.
Handlebar stem should be any 1" road bike stem made for a "threaded" fork. Try searching for "road quill stem". You handlebars should have the standard 25.4mm clamp diameter, though of course can't guarantee without looking at the bike.
The seatpost should have the size stamped into it just below the minimum insertion line. It will be something like 26.0 or 26.8. You do have to get the exact size. If it isn't marked, most shops should have a tool for measuring your frame to see what size it takes.
Now that I reread your question...are you looking to swap frames? The stem should fit any older road frame that has a threaded fork. The seatpost might fit another frame, but there are a lot of sizes. I wouldn't pick a frame based on whether your seatpost will fit it. They are cheap and readily available.