Understanding shoes for cycling and pedals for a beginner triathlete

Well, you will likely get a wide variety of answers since people have their own preferences.  Personally I use SPD-SL pedals/cleats ('clipless') and tri shoes for racing and warm weather training.  I also have a pair of road bike shoes I wear when the weather is colder but it's totally not necessary to have 2 pair of shoes. 

Some people do OK with cages that go on the pedals (plastic cage with straps) but personally I think clipless is the best way to go.  These are a bit of a misnomer since you are actually clipping in and attaching yourself to the pedal with the cleat as mentioned above.

They take a little practice to get used to clipping in and out but most people pick it up quickly.  Good luck to you in your training!

Clip on/Clip in refer to the "old school" toe cages and have been largely replaced by "clipless" pedals (which are really clip in...)

Cleats are the piece that attach to your shoes and engage the clipless pedals.  

Common types:
SPD:  Aka. Mountain bike style, also common on spin bikes.  Usually double sided with a small, metal cloverleaf shaped cleat.  
Shimano/Look/Keo:  Common on road bikes.  Usually single sided but weighted so they hang ready to engage with a larger, composite cleat.  
Speedplay:  Lollipop shaped pedals, double sided, with a large, metal, cleat with a receiving hole for the pedal.  

Speedplay pedals are popular because many of them offer adjustable "float."  I.e. the pedal/cleat combination allows some rotation side-to-side before disengaging.  I have Speedplays on my road bike.  I have SPD on my tri-bike.  For long races, I'll swap the Speedplays onto my tri bike.  I like either for triathlon because I'm less worried about running through transition with the metal cleats and I like the dual sided entry.  

Bike shoes fall into 3 categories:
Mountain bike shoes are typically designed for off-roads and are easier to walk in.  They may or may not have laces.  They may be woven nylon/canvas/composite but often have a rubberized sole with tread.  They usually accept SPD cleats and may or may not accept other cleats.  
Road shoes are typically used for road biking, usually with a strap instead of tie laces, but may have a Kevlar lace that locks in place.  These will generally accept most styles of pedals.  Designed for aerodynamics with smooth, stiff, hard, soles (often carbon fiber).  
Triathlon shoes are an offshoot of road shoes with a single velcro strap that holds opposite of road shoes.  They usually have some holes for water to drain out as they're intended for races where you don't dry off and don't wear socks.  In theory, they're intended to allow you to mount your bike barefoot with the shoes already attached to the pedals and you can slip your feet inside as you ride.  This skill takes lots of practice, and many races no longer allow this.  

I have road shoes even for tri's.  I never got good at putting my feet in while riding.  For anything longer than a sprint, I put socks on, put my shoes on, and run through transition with my shoes on.  (For sprints, I skip the socks, but still put my shoes on.)  In a HIM, you'll most likely be coming off the bike in the late morning when the sun has really heated up the asphalt.  *I* don't want to blister my feet just before running 13.1 miles so I don't run barefoot through transition.  

Shhh...

Here's my suggestion for this...if you normally put down your left foot, put the pedal and bike shoe on your RIGHT foot only.  Clip in with your right foot, but use a regular platform pedal and tennis shoe on your left foot.  You can start/stop, etc. and get comfortable with your right foot clipped, then switch...put the platform pedal back on your right side, put the clipless on your left, and practice starting/stopping as you clip in.  

While it may look silly to ride with two different shoes, it looks a lot sillier to be flopping around on the ground with a bike stuck between your legs (and yes...it WILL happen...)  

The pedals generally come with one set of cleats.  You can buy replacement cleats.  

Pedal fit is part of a standard bike fit as the cleats can often be adjusted on the shoe.