The Specialized will be fine in the Elite trainer.
The one problem you'll find is that knobby tires don't work well on trainers such as this. If you plan to primarily ride on the road I'd strongly suggest investing in some road tires. A cyclocross bike can take some fairly wide tires so I'd suggest running 28-32mm wide tires. They'll be MUCH more comfortable than regular 23-25mm tires, faster/more efficient than knobby cross tires and will also still allow you to ride on gravel roads. Trails that are too easy on a MTB suddenly become a real hoot on a cross or road bike! It's arguably a better way to improve your bike handling skills than riding a MTB on more technical trails.
I'm a big fan of Panaracer Pasela tires which can often be had for less than $30, have a nice ride, are quite durable and available in a variety of widths. If you want really nice tires check out Compass or Grand Bois. These are probably the nicest, cushiest road tires made but they also cost 2-3 times what the Paselas cost and won't last as long. Good tires can make a huge difference in how your bike rides. Finally, trainers tend to wear tires down much faster than regular road riding so I wouldn't go crazy investing in expensive tires if you plan to spend a lot of time on the trainer. They actually do make special tires just for use on trainers.
As for riding on the road, there is no easy way around that. Start on quieter streets or streets with bike lanes or wide shoulders. It really is just a matter of getting used to it. Riding with a group will also help.