3 dozen diapers and 6-ish covers, if you're using flats or prefolds or fitteds. If you're using pockets you'll need as many inserts as you have diapers, plus a few extra for nighttime or long car rides. If you're using all-in-ones, you'll probably want some doublers.
Detergent is a very personal thing. Most people do fine using the same detergent they use for their clothes. Some find that cheaper detergents like Sun work better. Others use Sportwash from Wal-Mart or order Allen's Naturally or Charlie's Soap from online.
What type of diaper is best is also a very personal thing; it's sort of like picking out a car. Flats and prefolds are cheaper and easier to care for but require more of a learning curve to use. Pockets and all-in-ones are the opposite, with fitteds somewhere in between. For covers, nylon and PUL are inexpensive, durable and waterproof, but not breathable. Wool and fleece are the opposite. Some folks prefer Velcro because it's easier and more adjustable, some prefer snaps because they're more durable and harder for the baby to remove, some prefer pull-ons because they're simpler (albeit messier at the end of a yucky change, and hard to get on over shoes).
Me, I like prefolds and front-snapping PUL covers except for newborns and potty training older toddlers. But that's just me!

P.S. GET A MINI-SHOWER or other diaper sprayer. Just trust me on this one!!!