There is no river in Copenhagen, you'll be on the canal side!
Loads of good places to eat without spending over the odds so it depends what you want. Jensons steak house (bofhus) are always good and they are dotted around the city. I'm useless with names but there's a lovely little Greek restaurant just along from Radhusenplaza in the city centre.
Nyhavn is very nice too but it can be expensive, all the menus are on view though so you can check things out before sitting down! The Rio Bravo on Stormgade (a 3 minute walk from Tivoli) is the place to go for the Danish dish of fried (think it's actually baked!) pork with potatoes and parsley sauce and you pay per person for as much as you can eat.
Of course, there are the many sausage stands ... the husband recommends the one outside the central train station across from Tivoli.