Your new character Ryder will leave the Milky Way galaxy on their 600-year journey to Andromeda in 2185.
That's the year that Mass Effect 2's events conclude, and that game's Lair of the Shadow Broker DLC takes place.
The following year, 2186, is when Mass Effect 2's Arrival DLC is set, when the Reapers actually invade the Milky Way, and when Mass Effect 3 begins.
By the time this happens, your ark ship (and four others) is already on its way to Andromeda, and its crew is in cryo-sleep. Hence, your characters will not have seen the Reaper war or how you chose to conclude it.
Or, if they are somehow informed after they awake, it already took place 600 years ago and was already resolved. Handy!
This places the main action of Andromeda somewhere in the region of 2785.