Installing the current version of Adobe Flash (11.2 r202 as of this post) in Fedora 17 is as simple as...
rpm -ivh http://linuxdownload.adobe.com/adobe-release/adobe-release-i386-1.0-1.noarch.rpm yum install flash-plugin
rpm -ivh http://linuxdownload.adobe.com/adobe-release/adobe-release-x86_64-1.0-1.noarch.rpm yum install flash-plugin
This installs Adobe's repository too, so flash will be updated along with your other software updates (eg
yum update). You can confirm the installation by navigating to
about:plugins in either Firefox or Chrome.