There are several threads in this forum on this very topic. Do a search -- the same shoes get recommended time and time again (reliable classics that work!). Since everyone's foot and stride are different, it's important that you try on the shoes and, if possible, walk a little on different surfaces (i.e. not just carpeting) before buying. If you have a good running store or walking shoes store near you, they should be able to assess your size and stride to help fit you with a brand and style that will suit you well. Make sure to break in your shoes before your trip and also to buy good wicking, arch supporting socks to bring on your trip. (We love Smartwool brand, but there are others that are very good, too.) Bring more than one pair of shoes so that you can switch during the day -- this will alleviate pressure point build up and help amazingly during your trip.
All that said, we love Keens (lace ups, mary janes, waterproof sandals). For us, that brand has worked consistently well over the years -- for plantar fasciitis, high arches, narrow feet, hip and knee problems, etc. Other DISers will jump in with what works best for them. Know your feet and their unique needs. Then try on different shoes to see what works best for you. Good luck!