I always found trying to clear dungeons on the first day that you are able too a fun challenge... then generally giving the screen a blank look as I realise I've failed for the tenth time trying to beat the annoying sub-boss. In other words generally try to get as far as you can in a dungeon on your first expedition into one. Although be warned, unless you have a certain item you need to use the little floating blue spectral butterfly checkpoints to get out of dodge, I mean, dungeons.
Oh and regularly dump the rubbish that you gather from enemies in the sales bin at the weapon shop, it will unlock new and usually stronger weapons and equipment you can buy for your characters. But make sure you're not dumping any side quest items by mistake or you could be in the annoying situation where you have to revisit old dungeons just to farm for particular items.
TRIAL OF THE DRAGON!
Edit: Also on your first playthrough you don't necessarily have to max your Social Links with everyone but should definitely try to get all your Courage, Understanding, etc. stats up to max by the end as it will speed along Social Links in New Game+. Additionally there is always the chance there may be Social Links you can't get access to due to your stats in the first playthrough that will then open up in the second one. Oh and go for all the Love routes. It ends up with some pretty funny dates and dialogue pieces.