For those of you longing to lace up the hiking boots or running shoes, Horsetooth Mountain Open Space offers several excellent trails to tickle your fancy. 
What are paddock boots and why do I need them?
Paddock boots ARE riding boots. They're pretty much any riding boot that isn't designated as a field boot or dress boot. They generally go about 6 inches up the leg (from the heel), and are more flexible than tall boots.
Since you're just starting out, hiking boots should be ok as long as they have a heel and aren't too large and clonky. I used to ride in an old pair of hiking boots, but sometimes I would have the issue that they wouldn't fit in the stirrup, they only fit in peacock irons since those are wider. But if you've decided to stick with riding, I would just go ahead and get paddock boots and half chaps. Should you decide to show, I would get the nicer half chaps that are made of grained leather (you should polish these occasionally) so then you don't have to get field boots.
I'm not sure what people told you to get for footwear, but you should get something close to the combination I have found lower down:
Boots:
http://www.doversaddlery.com/product.asp?pn=X1%2D38504&ss=paddock%20boots
Halfchaps:
http://www.doversaddlery.com/product.asp?pn=X1%2D35112A&ss=half%20chaps
You should get both in black, and make sure to take good care of them, i.e. knock the dirt off them after you leave the barn and polish them up about once a month.
Find a tack shop near you. Try stuff on, see what fits, and note what size you need in what brands. Then, go online, somewhere such as Ebay, to find exactly what fit in the store that's affordable.
You get what you pay for. Don't expect quality unless you're paying for quality. The boots that are listed above are about as cheap as you can get without getting synthetic leather.
If you decide not to show, I've listed below other half chaps you can get that aren't full-grained leather, and therefore much cheaper:
http://www.doversaddlery.com/product.asp?pn=X1%2D35319&ss=half%20chaps
http://www.doversaddlery.com/product.asp?pn=X1%2D35103&ss=half%20chaps
Notice the shoe she's wearing? Here's the link to it:
http://www.equestrian-tack.com/ariat1.html
Those are okay to wear, too. As long as there's a 1-inch defined heel.
EDIT:
Regular sneakers are a BAD idea. If the horse steps on your foot, you have no protection, and if you are riding and happen to fall off, your sneaker (with your foot in it) are more likely to slip through the stirrup and get caught. The tread is also too good on sneakers, so if you need to adjust the stirrup on your foot, it's going to be super-hard with a sneaker.
EDIT: Also, you should get halfchaps no matter what. When you go to try them on, wear whatever pants you're going to wear in lessons, i.e. jeans or breeches. I wouldn't invest in a pair of breeches just yet, but they do make riding more comfortable. Just make sure that if you're wearing jeans, you kind of twist the lower leg so the seam isn't on the inner part of your calf, and fold them over so they don't bunch up while you ride. Once you have the halfchaps on, it's hard to adjust your jeans (if your halfchaps fit right). The halfchaps should be snug but not too tight or too loose. They are meant to help you "stick" to the horse and are useless if they are too big.
EDIT: AND, you should NEVER get boots with a steel-reinforced toe! My friend wore them once. Her horse stepped on her foot, thus crushing the steel onto and into her foot, effectively amputating three of her toes.