Moscow summer
Moscow is a different city in the summer. From the beginning of June onwards the temperatures start to rise up into the low thirties - which, in a land-locked city this size, can feel pretty hot - and most Muscovites respond by trying to spend as much time out of the city as possible, either on holiday or, most weekends, at the dacha. It's also the time of year when most theaters close or go on tour, although provisions are made for the tourist season. June is still a good time to visit, though, and if you're planning a longer stay then the summer months are of course the best time to get out to the Golden Ring and the other attractions around Moscow.
If you're planning to travel out into the countryside then it's worth bringing insect repellant of some sort. Bear in mind also that, while you'll want to bring the right clothes to stay comfortable in the heat, urban Russians tend to dress a little more formally - which is not quite the same as respectably - than their Western counterparts and, although there are no strict rules for what to wear, the standard tourist uniform of baseball cap, T-shirt, shorts and trainers will not be considered endearing by the locals on any but the very young. The more soberly you dress the less you stand out, and this is generally an advantage.
Moscow spring
Spring is no unmitigated delight in Moscow. The end of winter means thaw, and thaw means slush and dirt, with temperatures hovering around 0 ºC often for several weeks in March. Somehow connected to this, it's also the time when things breakdown or get cut off most often. By April, however, the worst is over, the city is turning green, and the weather is relatively mild. This is probably the most popular time for tourists to visit the city, with the conditions just right for sightseeing and city life still in full swing before the mass summer exodus.
Early May is another time for nationwide revelry, with two public holidays - May 1st and Victory Day on May 9th - only a week apart and very little work done in between. If you want to see parades, fireworks and Russian street parties, then this is definitely the time to come.
Temperatures do not get much above 10 ºC before May, so you'll still need the equivalent of a woolen overcoat, and its worth bringing scarves and gloves just in case. Tough shoes or boots are also recommended to cope with the slush and mud and, although Moscow has a pretty dry climate, an umbrella is still an essential at this time of year.
Moscow autumn
Autumn comes on fast in Moscow, starting in mid-September. The mood in the city is one of slight resignation as everyone gets back to work after the excesses of the summer and gets ready for the onset of winter, but it's also the start of Moscow's cultural year, with a rush of premiers and openings after the summer hiatus to put some energy back into city life.
Late September may bring a week or more of what Russian's call "babe leto" - grandmother's summer - when Moscow's parks and boulevards are particularly beautiful, but by October the temperature will be dropping dramatically, and the first snows are soon on their way. They rarely settle at first so, as in spring, the city can become a little slushy, and tough shoes and an overcoat, plus hats, scarves and gloves are recommended.