@Nightshade
Good for you for looking ahead, and considering your options.
If you are in that delicious dreamy phase of first considering emigration, then I agree with Medea that you include other countries in your research.
However, if you are already sure that your country of choice is Switzerland, for example because you have close friends or family living here or because you belong to an organisation which is based in Switzerland, then you'll need to prepare for the specific conditions here.
If Switzerland is your aim, and since you appear (are we right in deducing this?) to have no formal qualification, I'd say that two central things you could do now, are to read, read, read (this forum has a wealth of information, especially if you plough your way through the sticky threads) and to find a way to learn a Swiss language. Choose one.
Naturally, your focus now is your new baby and your other two children, so I wish you health and enough sleep. Whenever you have a moment, in between, you could try to learn a bit of vocabulary.
I'd like to encourage you to perservere. Going to school while small children are at home is a challenge. If you can't manage that, it's okay, and realistic. You might try to find someone who lives near where you are now, of German, French or Italian mother tongue, perhaps another young parent, or an adopted grandparent, with whom you could practice the basics of the language.
Just in case you mentioned teaching only as one possible option amongst others, I'd like to add this: If you are not already set on teaching, you might consider nursing. I mention this because it is one area in which there is a long-term shortage of staff, here in Switzerland, so easy to get a job. Also a part-time job, which can be a good thing for a parent.
Nurses with foreign qualifications can't automatically transfer into the system here, but the standard procedures for recognition are known and set out. If this interests you, search for "nurse" or "nursing" on this forum. Nurses who do their training abroad anticipating the Swiss procedure, who reckon, from the start, with having to do those extra Swiss top-up courses once they arrive and who set aside the money for them, and of course who gain a command of the local Swiss language, can be up and running in their profession after a relatively short time (by which I mean about one to two years).