All the info is on the transperth website as mentioned by an earlier poster. It has a handy route finder, just type in where you are and where you want to go to, and it gives you the bus numbers, stops, connecting trains etc.
If you zoom in on where you live on Google maps, (make sure the transit option is ticked), you'll see little bus icons where the stops near you are, click on these and it will tell you the route numbers and the time of the next bus due. Look up the routes on transperth to see where they go.
The prepaid smart card is the way to go. 99% of regular commuters use it, just hold it to tagging reader and it beeps, do the same before you hop off and it deducts the fare automatically.
A dayrider ticket costs $9.30 after 9 or 10 in the morning, and allows you to travel on any bus, train, ferry etc. for the whole day. It's a good way to get to know the routes etc. to see if they'll suit you, especially at the weekends when they're quieter, just be aware the timetable is slightly different at the weekend. If there's more than 1 adult, get a familyrider at the weekend. Works out cheaper.
While you're on the buses, you can pick up paper copies of the local timetables. They're just below the driver in little plastic holders. Train timetables are available in the underground station.
Tripmate Perth is a brilliant app and gives you the timetables in an easy format. Handy for finding out when the next bus is due.
As far as the CBD is concerned, I think only the cats are free. The cheapest fare on regular buses is a two-stage ticket, within the one zone as far as I know.