It's definitely best to go by late spring or summer. Problem is that summer is very short and the weather unpredictable.
It can be rainy and cold (10-15° C) or sunny and super hot (35-40° C).
If you are looking for an extreme experience, try winter time (December - February).
I must admit that I haven't been to Siberia yet but a good friend of mine lives there. This section about Siberia is based on her information.
She ensures that you will survive even -40°C and you will be proud of it. White fluffy snowy fir trees as a bonus. :-)
The weather differs a lot from season to season. From +40°C in summer to -40°C in winter.
There is no special preparation needed. The medical system is good. There are farmacies all over.
However, if you go to wild nature, make sure you bring first aid kit and your regular pills/ medicine.
Beware of pickpockets and drunk people.
Maybe you will meet "gopniks". They are drunk and wearing Adidas training suits. Behave like you met a bear: don't look into their eyes, don't scream or shout, don't take pictures! Just keep on going slowly. ;-)
Public transport is good but it can be difficult, if you don't know any Russian.
Best use the train to see the cities.
For nature sights like Altai or Kamchatka, you can book a tour.
Or you get a rental car in one of the bigger cities. But driving yourself can be very exhausting since distances are very huge. 1.000km (!!!) (or 621 miles) of driving is average to get to the nature sights.
Flights can be quiet expensive. However, this is the biggest part of the budget. Two weeks should be about 1.200 EUR per Person (incl the flight).
Popular places in summer time might get crowded, e.g. lakes.
But you will be fine do things spontaneously.
great nature: Kamchatka region, Baikal lake, Atlai republic, national parks in Krasnoyarsk.
Cities: Novosibirsk, Krasnoyarsk, Vladivostok, Tomsk and more.
Trans- Siberian railroad: the railway goes through the entire country. It is a nice option to see Russian culture and nature, as well as to learn Russian swear words and the right way of drinking vodka.
Siberia is very big. So it is impossible to see everything by just one visit. You can try the following routes:
a) flight from Moscow to Novosibirsk. Take a train for a part of the Trans- Siberian railroad.
b) Flight to Vladivostok (or Khabarovsk), from there a train to Petropavlovsk-Kamchatsky to visit Kamchatka.
c) Flight to Tomsk. Then train and bus to do: Tomsk - Novosibirsk - Altai - Krasnoyarsk - Baikal.