Veri interesting experience
Panama vieja, is a bundle of history and a very nice experience; I highly recommend the visit. There's a visitor center, but if you drive must pay attention since there are not enough signs, start your archaeological experience there, you'll be amazed. Lots of pirat stories, treasures, battles and some humor too. Walk to the ruins, ok !! a bunch of old stones, but panamanians are very proud of them, go and you'll find out why.
Casco antiguo is not worth the visit, ugly, dirty and scary.
Plan a visit to Panama canal, worth the drive, there's much to see there, so take your time, 6 hrs top.
If you like mess and crwds, go to Colon dutty free zone, personally I was very disappointed. Not that cheap, didn't feel as safe as in any of the huge malls.
Visit Albrok mall, it may not be cheap for americans, but surely you'll find things hard to find in US.
Like beer?? try Balboa or Panameña, the rest are not good