You could just store them all online, yes, assuming you don't have massive amounts of photos. There's a file limit to the free Picasa Web Albums (although you could pay to up the amount). That's the major drawback to Picasa. (Although there's nothing stopping you from opening up multiple Google accounts.)
You can control the privacy of individual albums, yes. For example, you could make your wildlife album a Public album but leave everything else as a Private album. I believe there's also an "unlisted" option, so it won't show up if people search for it, but if someone is already at your albums, it will show up as a gallery.
You can also map your photos (or, if your camera geo-tags, so much the better), so you could keep track of which photos you took where and that will show up on a little Google Map within the album.
You might consider signing up for multiple photo sites and just playing around with a handful of photos on each and see which you prefer. I know people love Flickr and some of the other sites, while I personally like Picasa, so it really comes down to individual preference.
EDIT:
There are 3 options for albums, like I said earlier: public, private, and unlisted. Here are the official definitions of each, straight from Google:
Public: Anyone can view your public albums on your public gallery.
Unlisted: Only those who have the secret link can see your album.
Sign-in required to view: Only those you specify by name can view your photos.
