I'm a runner. I've been in the military for over 13 years, have been required to run for all of it, and have run recreationally for the last 2 years. I run about 20-30 miles a week, depending on what's going on (races). I'm also a medic and talk with a lot of Soldiers about running, injuries, rehab from injuries, and physical fitness in general.
I can tell you, without a doubt, that you can absolutely strengthen your feet and arches. And you should. Our feet were never meant to wear cushy shoes with arch supports. They were designed/evolved with muscles, bones, tendons, and ligaments, all meant to work without cushion or arch support. Furthermore, when we add built up heels and arches, we significantly affect our gait...the way we walk....which impairs our joints from properly cushioning our body. We aren't meant to walk with our heels striking the ground firmly. We're meant to gently walk on our midfoot, and run on our forefoot, cushioning our ankles, knees and hips.
I ran and walked on tradtional, built up running shoes for years, and had hip pain, foot pain, and shin splints. Three years ago, I found a pair of "flats." Basically, shoes without built up heels or much extra padding. I began walking on those and slowly transitioned to running. It was tough at first. My feet weren't used to the work. But now I run completely minimalist, basically in shoes that have no support and only protect my feet from rocks and such. I have NO pain, no shin splints, and have been injury free for years.
So...the reason your sandals made your feet hurt is for one of two reasons: They aren't the kind of sandals you should walk in, meaning that they don't have enough strapping to keep your feet from having to "grab" every time you step, OR, your feet are weak.
AL is right. VivoBarefoot makes an excellent minimalist shoe. So does Merrill. If you want strong feet (and GREAT looking calves, by the way), go for a minimalist shoe. Bonus...they're light and cool. :-)
If you do decide to go for a minimalist shoe, be sure that you take your time to transition. Read up on changing your gait from a heel strike to a midfoot strike. Don't put on too much mileage at first.
Good luck!