Cycle shoes & cleats...

The difference between tri shoes and road shoes is mostly that the tri shoes are easier to get on/off and are a bit better ventilated.  If you really care about shaving seconds off transition, you might do well to get tri shoes.  But, depending on how well-ventilated your tri shoes are, you might end up needing to get a separate pair of road shoes if you plan on training in cool/wet weather.

Whichever shoes you get, make sure they will be compatible with the pedal system you want.  All the different pedal systems (SPD, Look, Time, Speedplay, etc.) have their own kinds of cleats, so you need to make sure your shoe is set up with proper holes to screw your cleats into. 

Here is my basic explanation of the differences in shoes and pedals.

Shoes:
Tri shoes usually have 1 strap that closes on the opposite side of the shoe from a road shoe and have loops on the back of the shoe, primary differences are to make them easier to get in and out of "on the bike" instead of taking time to get in and out of "off of the bike".  The other difference is that they usually have drain holes to let water out of the bottom of the shoe.
None of these differences are that significant if you are not a "competitive" person, but they are advantages for Tri events and work fine on the road.

Difference in cost I find that the most you spend the stiffer the soal of the shoe (better power transfer) and the lighter weight the shoe is.  These are just general comments and findings.

Pedals/Cleats:
They all have there own advantages and disadvantages from weight, longevity, amount of float, easy and adjustability of getting out.  Etc.    MOST road and Tri shoes have 3 bolt style Cleats and will give you a better selection of shoes if you get pedals that use a 3 bolt style instead of 2.  Looks, Speedplay, Shimano's new SPD, I believe all use a 3 bolt style.

You could also consider sticking with "cages" until you know exactly what you want in a pedal/shoe.

Hope that helps,
Michael