Moscow

Since Moscow is the biggest transportation center in Russia and one of main the points of entry for the foreign tourists, it is a convenient starting point for exploring much of the European Russia. Even travelling through Moscow to Ukraine and some Caucasian and Central Asian countries Armenia, Azerbaijan, Uzbekistan etc. could be cheaper than direct flights from Europe/North America. Travel deals to Moscow are not rare and ticket prices are often pretty low within former USSR.

Saint Petersburg
13 different overnight trains leave Moscow for the 7-hour or thereabouts journey, arriving the next morning. You can either take the sleeper trains, which is a very interesting experience. Russians have a "train" culture, so it's very possible that you will meet your bunk mates and have a meal and drink with them. You can also take the high-speed Sapsan trains - they take about 4 hours, and the vistas rushing by are very lovely. You might even consider paying the extra money for a first class sleeper cabin which has two comfortable beds. Included in the price is a small snack for supper and breakfast. There is also an attendant for each carriage who is willing to make tea in classic metal and glass tea glasses. Very civilized way to travel.
Arkhangelskoye
One of the finest of Moscow Oblast's usadbas estates is only a short elektrichka ride away from Moscow and makes a fine day excursion.
Golden Ring
Old cities and towns rich in historical buildings, situated in the heartland of Muskovy Russia. There are many tourist companies organizing guided tours, but travellers with rudimentary knowledge of Cyrillic alphabet can do it independently. Many guidebooks are available in English.
Kubinka Tank Museum
(http://www.tankmuseum.ru/...) - One of finest armour collections in the world. About one hour west of the city. Access is restricted, visitors must apply for a permit (http://www.tankmuseum.ru/...), but it worth the bother for any self-respecting tank buff.
Leninskiye Gorki
An old country estate, expropriated by the Communist authorities after 1917 and used by V. Lenin as his country residence when he became ill. Large museum, although pretty decrepit now.
Borodino battlefield
This is the site of the famous Battle of Borodino. Museum and national historic site (http://www.borodino.ru/en...). Commuter trains from Belorussky Station; 2-3 daily, travel time about 2 hours.
Melikhovo (Chekhov's country house south of Moscow)
Sergiev Posad
Famous old Orthodox monastery Troitse-Sergieva Lavra. Commuter trains from Yaroslavsky Station, several daily; travel time about 1 1/2 hours.
Kolomna
A nice medieval town about 2 hrs from Moscow with a number of very interesting churches and monasteries
Yasnaya Polyana (Tolstoi's country house close to Tula)
New Jerusalem Resurrection Monastery
A monastery-fortress male, working with a number of museums inside and next to the walls: Wooden architecture museum, local history museum, Art and History museum etc. The monastery was founded in 1656 by Tzar Alexis II and Patriarch Nikon his "cell", a three-storey house stands in the park outside the monastery walls to resemble the original Jerusalem. The place is roughly between Novoierusalimskaya 15 min on foot and Istra 15 min by bus elektrichka stations, around 60 km from Moscow.
Savvino-Storozhevskiy monastery
A beautiful monastery with interesting history, closely connected to Russian Tzars. Commuter trains from Belorussky station to Zvenigorod, several daily; travel time about an hour, but the terminus of the elektrichka is 2 miles from the town, which boasts a number of historically significant churches itself, and about 3 miles from the monastery, which is on a nearby hill.
Dmitrov
A town about 65 km North from Moscow trains from Savelovsky station, several daily, travel time 1 1/2 hours, on Moscow Channel, with old churches, interesting sculptures in the streets and a number of museums
Snegiri
Settlement about 40 km from Moscow Volokolamskoe hwy that boasts a monument to the Defense of Moscow during WW2, with a good collection of tanks, and a museum. Trains from Rizhsky Station, several daily, travel time about an hour.