62 and want to start training for first triathlon

Welcome to BT and triathlon! It is ALWAYS a good time to start, regardless of age.

I'd suggest joining one of our open mentor groups--they are super for advice and support.

For a newcomer, I'd also recommend Sally Edwards's book Triathlons for Women and/or Jayne Williams's Slow Fat Triathlete (it's hilarious and very practical too). Good luck!

Welcome to BT Kat.  

Swim lessons are good.  

Not sure where you are, but as you look at bicycles, think about what bike riding you might do.  Different styles of bicycles have different advantages and disadvantages.  

Road/Triathlon bikes:  Lightweight frames and skinny tires make these the lightest and most efficient bikes.  Riding position may be uncomfortable, particularly if you only ride occasionally.  Those skinny, lightweight, high-pressure tires may be prone to flats and hard to fix if you do get a flat.  

Mountain bikes:  Heavy frames, knobby tires suspensions, and very low gearing make these go-anywhere bikes, but they're not very efficient with the big, soft, tires.  Riding position is more upright and therefore suited to casual riding.  

Hyrbrid/City/Fitness:  Kind of a mix between the two.  Frames are heavier than regular road bikes with slightly wider/heavier tires and usually a much wider range of gears.  A more upright riding position compared to road bikes.  A good all-around bike if you're going to ride casually.  

Recumbant:  These are the funky looking bikes where you lean back rather than forward.  Good if you have back or wrist problems.  

I started with a 10-speed back in high school.  Years later, I got tired of fixing flats and dodging glass and gravel while riding around the neighborhood with my kids and got a mountain bike.  A few years later, I did my first triathlon and bought another road bike and then a tri-bike.  I still have the mountain bike, and I ride it to the store because in town, I want to sit up and be seen, not worry about glass, carry stuff I buy at the store (I have bags on it) and ride in tennis shoes rather than my bike shoes.  If I were only going to have one bike and ride casually, I'd get a fitness bike rather than a road bike.  This has the added bonus that fitness bikes start at about $250 (sporting goods stores) where entry-level road bikes are about $800 (not including pedals, shoes, pump, repair kit, pouch, helmet, cycling computer, etc.)  

So...

Would you ride your bike to the store to get a few things?
Would you ride around your neighborhood?
Would you ride your bike 2 miles to go to the park or a restaurant or show (instead of driving)?  
Would you ride on busy streets?  
Would you want to carry things when you ride?  
Do you want to go fast, fast fast?