Summers in Novosibirsk are typically warm, but sometimes cold and rainy. Winters can be really cold with temperatures as low as -40 degrees celsius.
Take a train from Moscow or from Saint-Petersburg (about 2-3 days). You'll arrive at the Trans-Siberian Railway's largest station - Vokzal-Glavny (russian "Вокзал-Главный"). So close to this station you will see the Metro (subway) - Garina Mikhailovskogo station (russian "Гарина Михайловского"). There is 13 metro stations.
Or fly to Novosibirsk Tolmachevo Airport, with S7 Airlines or other one (Aeroflot, Transaero an so on). You can use 2 buses, the first one to Vokzal-Glavny, bus number 111, or you can use bus number 112 to Bus terminal or Avtovokzal (russian "Автовокзал"). You will go about a half-hour till one of these stations.
There is a lot of public transport in this city. Everywhere you will pay from 16 till 25 rubles (it depends from your change of transport). Taxis, metro, buses, street railways or trams and so on. Most citizens can't speak even basic English (although almost all of them were taught English in schools). Very often it depends from some english accents of tourists, I mean, misunderstanding each other.
The Metro  is cheap but only runs through the city center and just past the Ob River. Still, for the visitor, you can get to many major destinations from its six major stops, including the main square, Lenin Square, on Krasny Prospekt (Red Avenue).
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.
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.
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.
Many cafés and restaurants offer paid (and free) Wi-Fi all over the city.
There are some areas in the city to shun. Pervomayskiy rayon and far-off areas of Leninskiy and Kirovskiy rayons should be avoided at any time.
The safest place is the center (nearby Lenin Square), stay there and you will be all right.