Given the bike has vertical dropouts (where the rear wheel fits in), there are some challenges with this bike. Chain tension in a fixed gear bike, because there is no derailleur, is done by moving the rear wheel back. You can't do that in a bike with a vertical dropout unless you buy a special rear wheel with what's called an eccentric hub or replace the bottom bracket with an eccentric bottom bracket. These are expensive options. An eccentric hub will run you around $200, That's for the hub only. You'll pay more to have it built into a wheel. Once that's done, the only other consideration is replacing your crankarms if they're long (say 175 mm). Because you can't coast on a fixed gear, if you're cornering, you can catch a pedal if the crankarms are long.

Before you consider turning this bike to a fixed gear and drop that kind of coin, consider if you'd be interested in going in one gear all the time, which is what you have to do if you're on a fixed gear. Try that on your Tourna but setting a gear that works for you and keeping it there, without shifting. If that's okay, my recommendation is to look around for an older road bike, one with horizontal dropouts - check Sheldon Brown's page that duppy posted to see what I mean. This is the time of year that a lot of bikes are coming up on Craigslist and chances are you could snag one for not too much. Then buy a decent quality fixed rear wheel, which will likely actually cost less than the hub. Then you'll have a better bike for riding fixed for less than if you convert your Tourna and you'll have two bikes, to ride whenever you feel like doing one or the other. Conversions on modern road bikes can ge challenging, relatively more expensive, and aren't the best option, IMO.

And welcome