How can I find out the storage type of an EC2 instance's root volume (SSD vs. magnetic)?

If you select your instance in the web console you can see the volume ID of the instance (it begins with "vol-"), clicking on that will bring you to the volume in the "Volumes" section of the EC2 console. In the storage type column it will tell you whether the volume is "magnetic" (old volumes), GP2 (new ssd) or PIOPS.

Edit: Additionally, you should be able to tell if you run (using the AWS CLI)

aws ec2 describe-instances

davidgo is right. You can measure the speed. This hint is from https://4sysops.com/archives/how-to-check-if-your-ec2-instance-uses-ssd/

Example:

sudo ioping -R /dev/xvda1 -c 1000

Replace xvda1 with your device/filesystem. ioping stats for SSD should show way much faster rates.