Don't do it in late summer. There could be very hot and stuffy in summer, especially in large cities.
The best time to visit central Russia is May. June is usually ok too. If you do it in late June visit St.Petersburg and enjoy white nights.
Moscow is nice if you never was there before.
If you're kind of adventurous man fly deep into SIberia to Baikal lake, or even farther to Kamchatka.