If you have only 2 weeks - 3 days in Moscow = 11-10 days, then forget about European part of Russia and go straight to Siberia. I don't know why everybody suggests TSR, but personally it is very dull to sit in train for a week.
First try to feel Moscow life rhythm. You will see later how it is different in Siberia.
Then I suggest you to fly from Moscow to Yekaterinburg and move from there by train. IMHO Siberia gem is in its boundless territory with some scent of wilderness. You can feel it only by traveling by train or car. In Siberia you can feel that Moscow is far away from that region (no wonder that fugitive serfs ran away to there to start a new life).
Try to visit Tomsk, Omsk. Visit Kemerovo to see industrial Siberia, though I think that it is the worst city in Russia 8) Then go to Altai. Altai is a must. Krasnoyarsk is a must too(Yenisei is a mighty Siberian river). From Krasnoyarsk fly back to Moscow. I don't think that you would be able to see Baikal due to lack of time. BTW Altai also needs extra time, but if you traveling around why don't visit it even for a while.
Keep in mind that spring is time when ticks wake up from winter sleep. Siberia was always in front lines of tick encephalitis infections in Russia.