I highly recommend both of the games for the PS2 made by Team Ico. Their debut title, Ico, has a heavy focus on puzzle solving that somewhat resembles that of the Zelda series. Their second title (and my personal favorite game of all time), Shadow of the Colossus, is centered around exploration and combat. Both games are masterpieces (my opinion of course) and I think a Zelda fan would feel right at home with either title.
EDIT: S'more great series that are a little different than Zelda but are nonetheless amazing are Fire Emblem, Kingdom Hearts, Final Fantasy, and Ninja Gaiden. The last one is a bit of a reach, but I was just remembering how cool those games are while I was listing stuff.