About the restaurant, i’d say Lido. It’s in the area of Elizabete’s iela, but you can find information about it on google. Typical latvian cuisine at good prices.

About the one unmissable thing to see… Hmmm, I’d say the market, st. Peter’s Basilica (there is an elevator to the top and the view is stunning), the house of the blackheads(these three are very close to one another), the monument to freedom in Brivibas Bulvaris (brivibas is an avenue) and while you’re there take a walk in the park between the Opera and the freedom monument.
Of course that while you’re going from each place that i said to another, you’ll see a lot of beautifull things and the old part of Riga is not that big so you can see most of it.

Hope to have helped, Carlos

First – Lido isn’t just in Dzirnavu iela (mentioned as Elizabetes iela in the first answer), but there are several Lidos all around Riga. And well, about the “traditional” Latvian food… ok, I can’t disagree, but it’s not really what Latvians usually eat…(Maybe on some special occasions). My personal favourite is the Kiploku Krogs (Garlic Pub) in the Old town, but it’s difficult to find (Maza Pils street, if you’re interested).
Second – well, the Old Riga is the best place for sightseeing :) But, if the weather is nice, you can also visit the Open-air museum. It’s interesting and relaxing, even the locals (like me) like it :) But well, yeah, better stay in the Old Riga, if you don’t want to get lost (actually Riga is rather tourist-unfriendly…). I don’t know about the night clubs (mentioned in another answer), they’re exactly the same as in other countries (stupid, with stupid music and a lot of stupid people) (it’s the right place to be if you’re stupid too…)
Third – museums rule. Visit them. (I just have never noticed where is the information in English… if there is any…)

Ok, just look around you and enjoy the Old town ^_