I did my certificates and ratings in the order:
Private, ASEL (Airplane Single-Engine Land) (1987)
Instrument (1988)
Commercial, ASEL (1989)
CFI (1991)
CFII (1991)
Commercial, AMEL (Airplane Multiengine Land) (1994)
MEI (1995)
ATP, ASEL and AMEL (1996)
I paid for my multi commercial, MEI and ATP using income from my flight instructor job. The CFI job didn't pay very well. I once calculated that I had to give 32 hours of dual instruction to pay for each hour of training in the twin (a Beech Baron).
After all that flight instructing, the MEI was very simple, probably the easiest checkride I've ever taken.
Instructing in a twin (especially a high-performance one, like the Baron) will keep you busy. It also has the potential to be quite hazardous. I'd recommend to any future MEI that they spend a couple of hundred hours teaching in single-engine aircraft prior to doing any multiengine instruction. (In over 9000 hours of flying, the absolute worst situation I've ever been in was after I had an engine failure, in a twin, while instructing. We came that close >< to going into the trees.)
Unfortunately, there's relatively few flight schools where you will be able to do a lot of multiengine instruction. One way to get around this problem is to flight instruct until you reach Part 135 minimums (1200 TT), then get a job flying freight in a twin. As a freight pilot, I was able to log over 100 hours a month multiengine time. The best I ever did as a CFI was 50-60 hours of multi time a month, and that was at a very busy flight school that specialized in multiengine training.
One comment: I WOULD suggest you get your CFII immediately after you finish your CFI. Compared to the CFI, the CFII is very straightforward (I did mine in under a week). This will allow you to work with instrument students in addition to private and commercial applicants.
Best wishes,
FFFI