The first thing you need to know are the basic types of diapers. The ones you used 17 years ago are prefolds and covers. The prefold can either be put on the baby and pinned or Snappied on, or just trifold the prefold and lay it in the cover. Similar to those are systems like the Flips, which use their own insert of some sort and a cover. Same basic idea as trifolding a prefold and laying in. With either, you can reuse the cover if it didn't get pooped on.
At the other end of the spectrum is the all-in-one. It's one piece, with absorbent and cover all sewn together. Just lay baby on the diaper and fasten (velcro or snaps). These can be more difficult to get clean and take longer to dry because one side of the absorbent layer is waterproof and water and air can't flow through.
In between are pocket diapers. These are two pieces, a shell and insert. The shell is basically a cover, with an extra layer. There is a pocket that lets you stuff the insert in between the layers. To use them, you stuff the pocket with whatever you're using for absorbency, then use it like an all in one. When you take the diaper off, un-stuff the pocket and throw it all in the wash. Some people stuff as they are changing the diapers. I prefer to stuff as they come out of the wash, which lets us use them like AIOs. I like pockets because I can change the absorbency as DD's needs change, and they wash up easily and dry quickly.
There are some other types of diapers but those are the three big categories.
Diapers (covers, pockets, and AIOs) either have snaps or velcro-like closure. Personally, I prefer snaps. Velcro (often called Aplix, another brand name for hook and loop) style can become pilly and if you don't use the laundry tabs, you can end up with a long diaper chain when you pull it out of the laundry. Velcro also wears out and needs to be replaced. I've heard that velcro provides a better fit for little babies, and snaps for toddlers. I only have experience with CDing a toddler so far. When I first started CDing, I bought 2 snaps and 2 Aplix to try and see which I liked better.
The other decision to make is one-size or sized. A one-size system has snaps on the front of the diaper to adjust the rise. Some people prefer a sized system because they feel it gives a better fit.
As for how many, for a newborn, if you're washing every 2-3 days then 30ish.
