Any recommendations as to where?
Firstly, you should choose the trains. For now, you have such options:
- Direct train 020Щ(sch) "Vostok" ("East")
Moscow (departure at 23:55 PM every Saturday) - Beijing (arrival at l5:32 AM every Saturday), do not goes through Mongolia.
- Direct train 004З
Moscow (departure at 21:35 PM every Tuesday) - Beijing ( arrival at 14:04 PM every Monday), goes through Mongolia.
- Direct Carriages from Novosibirsk, Irkutsk or Chita (schedule is on link above). In this case you can travel from Moscow to any of this cities, do some break there, and use the carriage ticket. If I were you, I chose the Novosibirsk to visit beautiful city or Irkutsk to visit Baikal lake.
Since I'd like to travel in either a Spalniy or Kupe compartment, it appears that I would need to book well in advance. How can I best book my tickets if I plan to break my journey into 3 or 4 legs?
You can book the tickets up to 60 days from your departure date. Also during your trip you can pause your ticket at any station (if your passport and visa allows this). In this situation your must get stamps on train or on station you got off from train. You can do this during 60 days from your departure date.
Also you can continue your ticket up to two times for 60 days, but this can be done only because illness or something more important.
Note that you will lose some part of ticket price in case of breaking your ticket - you'll have to buy new ticket part (not very expensive, however).
Are there different visa requirements if I plan to stop off in, say, Mongolia for a few days, rather than making the trip in one stretch? (See also: What is the best way to obtain visas for the trans-siberian/trans-mongolian railway?)
I think you should do the break on Russian territory, just because it is simpler to maintain.