Hedge,
You will be ok with running shoes, just like you'd be ok with a mountain bike. But, a faster bike and bike shoes/cleats will make you faster/more efficient. If you were riding 2 miles to work in Amsterdam, I'd say stick with your street shoes and an upright bike. But you are talking about putting in some good miles, so my recommendations are based on the distances/terrain you're talking about.
Links below are just for illustration of the terms, not necessarily for items I'd recommend.
Terminology:
Toe clips (the old school plastic or metal cage with a fabric strap that holds any shoe in place on the pedal). Used with "regular"
platform pedals and any kind of shoe.
Clipless pedals: Confusing term, but these are the kind that
don't look like a pedal, but are some device that locks into a
cleat on the bottom of a cycling shoe to hold your foot in place. Certain pedal systems work with certain shoes, so you need to check for compatability. They are called clipless (despite the fact that you "clip" into them, because you don't need the old school toe clips. They will require more maintenance than platform pedals, and if you don't have them set up right, some people have knee problems from poor fit.
The benefit of cycling shoes/cleats/clipless pedals comes from 1) Cycling shoes have very rigid soles, which let you transfer more power to the pedals and not waste it with the flexing of the soles of the shoes 2) fixed attachment to the pedal allows for more efficient pedal stroke. Instead of just pushing down on the pedals like on a platform pedal, being clipped in and using good technique lets you push the pedal forward at the top of the stroke, and pull it backward at the bottom of the stroke, as well as unweighting when lifting your foot and pushing down during the downstroke. So, it lets you use more muscles and transfer power better, meaning more speed/efficiency and less fatigue.
If you don't want to spend the money on a full setup now (why should you, if you are just testing the waters?), I would spend a few dollars for some old school toe clips on platform pedals. You can start with your regular shoes. If you want to switch from running shoes, you could get some "walkable" bike shoes which have fairly rigid soles, but have a recessed area for the cleat, so they are relatively easy to walk in. Something like
these, but get something cheaper/on sale. You can wear them with toe clips, but without cleats to start, which will give you some benefit. If you start doing lots of long rides, you'll probably make the switch to clipless at some point. And you'll probably fall a few times when stopping before you remember to unclip :-)