lenin square (city center)
See the Lenin's square with a grey-color monument of Lenin left from Soviet era standing right in front of the Opera and Ballet House which, by the way, is beautiful at night. It is also the largest opera house in Russia. There's also the exact geographical center of former Soviet Union just south of Lenin Square, marked by the Chapel of St. Nicholas, originally built in 1915 to commemorate 300 years of Romanov rule, destroyed in 1930 and rebuilt in 1993. In winter residents build an ice town on the Ob River, and in summer you can go to the beach. Near the city center on Krasny Prospekt,you can also see some excellent works in the Novosibirsk Art Museum and Local Studies Museum. You could also consider the Novosibirsk Regional Puppet Theater over 70 years old in an interesting historic building, Novosibirsk Circus, and several other theaters.As in every Russian city, there are multiple monuments to heroes of the Great Patriotic War WWII, including an eternal flame guarded at special ceremonies by youths with replica Kalashnikovs. Those are standard touristic places, but you should definitely make friends with locals to see the "best" parts of the city.
akademgorodok
A visit to Akademgorodok, built in the 1950s as a mecca for academics but suffering from neglect in the post-Soviet years, is still pleasant as a day trip. This formerly elite institution is located about 30 KM south of the city and features several museums and nice, tree-lined streets.
While you're in the area, you might also visit one of several nice beaches on the Ob Sea, the large body of water created by a dam on the Ob River.
Also south of the city, about 2 km north of Akademgorodok, is the open-air West Siberian Rail Museum. There are about 90 historical trains and rail cars there.
novosibirsk zoo
The zoo is one of the largest in Russia after Novosibirsk had a really tiny one for decades until 90-s. Much money have been invested in the zoo during "prosperous 00-s", thanks to the zoo's devoted director Shilo, a member of the bureaaucrat United Russia party. The zoo is very well-attended and one of the best places to visit that Novosibisk can offer. (http://www.zoonovosib.ru/)