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I was in Riga this summer. There are some interesting sights there. First I would recommend just strolling around the beautiful old town. It isn't too big, but there are a lot of small restaurants, cafes, and bars, where you can have a drink.
Unfortunately, it can happen that there are a lot of English stag parties that are sometimes a little bit annoying, especially if they are heavily drunk.
Worth a visit is the cathedral, particularly because it is possible to ascend to a platform from which you have a superb view over the whole old town and also most parts of the newer parts of Riga.
Next to the cathedral there is a square, surrounded by some beautiful old buildings.
Just on the other side of this square there is a museum that explains the occupation of Latvia. In Latvian it is called Latvijas OkupΔcijas muzejs.
As a plus, the entry is for free.
On the other side of the old town, there is a freedom monument, that is also worth a visit if you're interested in history. It commemorates the day Latvia became independent of Russia (after WW1 and not in the 1990s). Exactly, it was built to mark the 17th anniversary of Latvia's declaration of independence. This monument also marks the border between the old town and the city center. Usually it is surrounded by red and white flowers. This is because in the Soviet area, flowers were banned!
Next to this freedom monument there is also a nice park in which you can chill and hang around.
If you're an architect, I would also recommend to go to the Swedbank headquarters. It is the first skyskraper that was built in Latvia (2004). Although there is a cafe on top of the tower, you can't go in there if you don't have a friend that is working at Swedbank.
A little bit outside of the old town there is a TV tower. It looks a little bit like the Eiffel tower and it is the tallest building in all Baltic states. There is also a platform for visitors with a 360 degree panorama.
If you're interested in old Soviet style buildings, I would go to take a look at the academy of sciences. The local population is not keen on this building, because it was built by a Russian architect and it was a gift from the Soviets to mark the western bounds of their empire.
I also thought that the market place is a very interesting place to visit. It is located in two huge old halls where airships were built. It used to be one of the most modern and largest markets in 1930s. Today it is still very impressive, and you will also be able to cover your second question there, i.e. there is a lot of food available.
Some very famous buildings are the houses of the three brothers.
And also the Orthodox cathedral is worth a visit.
And last but not least, you have to try the Black Balsam, a traditional liqueur.
All images are from Wikipedia.