custermustache is right, and work101 is too kind.
Even though you have a Suntour freewheel, it's an Ultra, so it's spaced the same as the Shimano freewheel.
This is good.
For a conversion with the least fuss, 7-sp:
Shimano RSX 7-sp (metal, repairable) or Shimano Sora (lighter, not repairable).
Simply swap in and go, which I've done several times when someone wanted "modern" with low expense.
I'd try the NR RD, just in case it works, but a Shimano 7-sp RD is really the way to go.
When you "upgrade" to STI, generally it's easier to match 3 things: cassette/freewheel=RD=shifters.
I like this method for any speed. Other options work. This method has fewer obstacles.
Last edited by RobbieTunes; 09-07-15 at 07:41 AM.