Seoul Tower will give you an amazing view of vast swathes of Seoul.
For history, Gyeongbukgung, Gwanghwamun, or Namdaemun are probably the most famous. The first is the palace of the last dynasty of Korea. Gwanghwamun is a restored gate from the old city wall that has a historic reenactment of the changing of the guards, and is sharply contrasted with the ultramodern feel of the surrounding cityscape there. Namdaemun is another gate of the city and is considered the top national treasure. There's also an open air market near the gate that's one of the oldest running markets in the country.
I can't recommend specific restaurants since I haven't lived there in years, but if you're looking for a taste of daily life, I'd recommend trying barbecue that's done right at your table. If you're old enough to drink, I'd combine that with soju to have the meal the popular way!
Like any city, there's plenty to see (and eat). These are some of the bigger attractions to the city, though your actual interests may influence what you ultimately want to do.
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