It seems pretty time-consuming to have to check dozens of asteroid fields to find the type of ore I want. Is it possible to see what types of ore are available in asteroid fields without actually visiting them?
Yes, it's not just possible but also very easy.
Well, assuming some macrominer hasn't stripped the belt on you already. 8 )
If you practice with your ship scanner, I believe you can scan for asteroids. Also, it's important to note that you can usually tell what kind of ore you will find in the system based on security status and the ruling faction (which type of pirates or which empire). Look at the information on Grismar's wiki to help pick which system(s) you want to mine.
My mining strategy was to find a system or group of out-of-the-way systems where I liked to mine, keep a Covert Ops ship there along with my Hulk and a hauler, and then find a good belt with the Covert Ops. Or better yet, have a buddy come along to cover you when you're mining, as it's a dangerous business with all the pirates out there.
For general mining information, look no further than Halada's mining guide, as it's the premier mining guide for EVE. He's done all the math for you and tells you basic mining strategies.