To expand on what SqrlCub said- turn based fighting is when you select an attack command (typically from a menu) and then watch as it is executed- a la the Final Fantasies. You then watch as the enemies take their turn attacking you. The alternative is “real time” fighting (like Zelda, Metroid, etc…) where you simply press the proper attack button and it instantly occurs. I find the turn based stuff to be kinda bland- the game has to have a real good storyline to keep me involved with one of them.
Secret of Mana is probably the most similar to the SNES Zelda in gameplay, but it’s MUCH more complex, and has more of an RPG factor–more interaction with characters, more towns and areas to explore. I’d recommend that as a good transition to…
Final Fantasy III. An amazing game in all respects. Easy to sink 40-60 hours into it on your first go at it, and still not find everything there is to find.
Or…
Chrono Trigger. The game has something like 17 different endings. I haven’t even come CLOSE to finding them all. The game is absolutely great, and it obviously has amazing replay factor.
Others recommended Earthbound, which is a great game–very cute, but not incredibly easy.
Lufia 2 is a wonderful game for SNES which didn’t get much publicity. It involves turn-based fighting, a decent storyline, and dungeons with lots of strategy puzzles. A nice combination–it works all parts of the brain. The first Lufia wasn’t nearly as good.
And for the person who mentioned that they were getting a Playstation 2 for Chrono Cross–it’s out now for the original Playstation. A friend of mine has it, and he says it’s awesome.
Right now at JCPenney, you can get a Dreamcast with two games for $159.00 (online or in the store, I verified both today). Then use coupon code ‘cr8’ to get another 30% off that [or anything over 150 bucks], then if you do it before Sept 1st, you get $50 rebate from Sega.
YOur price comes down quite a bit, & you can get some games for it like Crazy taxi which is a lot of fun. Silver is a lot like Zelda & very beautiful.
I got the free Bleem from Bleem.com it runs my Playstation FF8 game disks that I got for $17.00 used from BB. Shucks, man, getting used RPG or adventure games, PSX & play on your PC saves you a load of dough.
Chrono Trigger, most definately. It’s my favorite game of all time. Also, buy a playstation for Chrono Cross, which is also AMAZING. I’m about 14 hours into it, and it’s the best damn playstation game ever made. Maybe it’s just nostalgia, but Chrono Trigger is still the best game ever 
Regardless, the sequel certainly lives up to(although doesn’t surpass) the quality of the original.
I LOVE Secret of Manna!
Zelda was a pain in the ass, because it was so hard to maneuver, and you’d fall off a cliff and have to start all over.
I reccomend Breath of Fire and Breath of Fire II.