mtlbiker is right on the money. Fixed gears are not for novice cyclists, and really are only good 'cross' training for competitive cycists. Even then there are coaches who strongly advise against fixed gears unless you're training for track racing. I train a lot on fixed gear bikes and do a little track racing now and then (I suck at it).
Really, the reason they are attempting to cajole you into a 'fixie' is for style points. They're pretty much a fad these days, and there are a number of manufacturers making fixed gear/single speed frames and complete bikes.
If you want to look at it as a mode of transportation, a single speed is nice for its simplicity, but I wouldn't use it as 'cross training' for exactly why mtlbiker states, you'll wipe out your knees on a hill.
A fixed gear also requires some finesse and technique that you don't have yet. You have to remember, on a fixed gear, you _cannot_ stop pedaling. One of my first times out I was spinning along nicely, rolled up to a stop sign, and attempted to stop quickly, the group was highly entertained - you had to be there, even I think it was hilarious now.
Also, as a novice, do _not_ let anyone talk you into a fixed gear without brakes. They'll tell you that you can stop by using your legs, wrong..... as a novice, you'll need both front and rear. If you're riding around in the city on a fixed, there's no way you'll be able to stop quickly enough for a car pulling out in front of you just using your legs, and that's a sure way to severely pull a muscle or even tear a ligament depending on where your legs are when you try to stop. Riding without brakes in the city is just plain stupid in my book.
You don't need to spend alot of money. One thing you can do is buy a used low-end road bike and convert it. All it takes is removing the derailleurs and shifter cables, replacing the rear wheel with a single speed freewheel, and possibly adding a chain tensioner if the bike has vertical dropouts. You can do most of this yourself, though you might want some help if the bike has integrated shifter/brake levers.
After a while, you might want to try riding fixed, but I would get used to the single speed first. In that case, all you need to do is unscrew the freewheel and screw on a fixed cog in its place, but remember, don't play in traffic.