Bud:
Type of corker is the real answer. With a floor corker, any size will do, but I would recommend #9s. I haven't used a hand corker much, but some people report trouble inserting #9s, so they prefer #8s. BTW, I suspect these people have the squeeze together type and don't have the hand strength to compress the #9 corks, but I don't really know for sure. BTW #2, I'm a big guy, and I HATE the squeeze together type of hand corker. I only DISLIKE the other hand corkers.
Summary of sizes:
#7 - rare, but I do not recommend for anybody
#8 - good for most cork finish bottles
#9 - better for cork finish bottles, but may be a problem with hand corkers
If you must cork screw cap wine bottles, use #9s if at all possible. IMO, the mouth opening on most screw caps is slightly larger than a cork finish bottle.
Length of cork...if planning LONG storage use 1.75" (1 3/4") corks. For short storage use 1.5" (1 1/2"). What is long storage? Well I just opened a bottle that was bottled in April 2007 with a 1.5" cork. Just fine. Very nice in fact. Enjoyed it immensely. Gotta make another one of those kits someday.
I mostly use #9 x 1.5", but will sometimes use #9 x 1.75" if I really expect LONG term storage.
Tapered corks? 6 months? Personally, I would try to avoid that.
Steve