Completely clueless and motivated

Welcome!!!!
i would imagine that walking 2 pitbulls that distance every day provide a bit of a challenge at times!
if youre comfortable with the running/walking program you've started, just stick with it and you'll be racking up more miles than you ever thought in no time!  just follow their recommendations for building up times and distances.
most people will say, and i can agree, that you will be fine to use a  mountain bike for  a few races.  see if you like triathlon first before you go out and search for a new or used bike and the frame types and components and sizing.  building a decent fitness base is more important to do first.  and the swim, as long as your comfortable and swimming isnt anything completely new to you, you're better off than lots of other folks when they've decided to take up this sport. 
im in my second year of tri's. i have a little know-how, but im not a wizard. there's so many seasoned people on these forums who can motivate and educate you.  use it up!
one thing i can guarantee is that the races are totaly fun!
best wishes to you!

If you can join a Masters Swim group (with a good coach) at least once a week you should be able to increase your speed and endurance pretty well over the course of 6 months.  If you stick with it and find time to swim 2-3 times a week, the fitness benefits will pay off dividends.

I've used a mountain bike on my first few races. If you don't mind getting passed, you'll do fine on it.  Just get in at least 1-2 rides a week (or do some spinning at the gym).  It will help your running too.

Can you work your way up to jogging with your dogs?  Go a few minutes walking, a few minutes jogging.  Each weekend, take one hour to do a "long" run. Start out with a half mile and work your way up to 6 miles, or whatever feels comfortable.  If you can run 6 miles, you can easily do a 5K.

Having the spare time is the hard part.  If you can work in your runs with your dogwalking and find a few extra hours each week to swim and bike you should be able to do it.  Maybe eventually you could get the kids involved?

This user's post has been ignored.

Welcome to BT!  Yes, you CAN do it. 

You'll find people riding all kinds of bicycles at races.  Mountain bikes, hybrids, beach cruisers, road bikes, tri-bikes.  Two wheels, plugged bar-ends (for safety), and functional brakes get you about 95% of the way there.  I can just about guarantee you won't have the oldest, heaviest, rustiest, or slowest bike there.  You'll need a bike helmet (not like you'd ride a bike without a helmet...right?? ) 

Get back in the pool.  A few lessons might help (if nothing else, to get you to the pool regularly). 

You can look in the mentor forum for an open mentor group.  That'll give you a few on-line "buddies" to help you along the way.