I've been riding my Gunnar Street Dog, Fixed Gear, for twelve years now. Love it. Maybe it's best if I put it into "bullet points":

-Like most adults who work for a living, I simply don't get to ride as often as I would like. A fixed gear goes a long way toward giving me a better work out, whether
it is an evening ride, or an eleven mile commute to work.

-It's fun to ride. Yes, there is a learning curve, but if you are already a cyclist, you should pick up on it fairly quickly. You will make some muscles sore that never got
sore before, but this passes. Hills? Yes, there might be some that will stop you, but I regularly hammer up the hills in my area. Just stand, and put some butt muscle into it.
All of a sudden it feels as though you are getting a push from some unseen force. You will need to consider what gear would work best for you. Mine is 42x15, which gives me
a 75 inch gear. Some like a lower gear. I would probably go lower, if I rode into the city on a daily basis.

-The late Sheldon Brown spoke of finding a new joy in cycling, when you start riding fixed. He was right, I think.

-It's fun to ride. (yep, I said it twice)

One thing: Do not fall into the nonsense about "you don't need brakes on a fixie!" Leave that smurf for the 20-somethings, who seem to feel they're making some sort
of political statement. Riding brakeless on the road is very foolish, and illegal. One of the things I love about being over 50, is not having to care what anyone thinks.

If you decide to do this, give yourself time. Ride around your neighborhood, practice starting and stopping. When you feel confident enough, then take it out into traffic.