This is actually quite a tricky question, which will depend on a lot of aspects. Let me try to give you some ideas on where to get the information you are looking for.
Swiss taxes are quite complicated. As you are taxed on a federal, a cantonal (state) and a communal level, your taxes will vary a lot. Even if you live in the city itself or at a neighbouring town in the agglomeration, this can change a lot.
To complicate matters the taxes are also dependent on your income (as in there is not one percentage for everyone) and deductions you can make. These may also depend on the location you live in. As a foreigner you will most probably be taxed at a fixed rate on your income, however you can ask for it to be recalculated to the more exact rate.
Have a look at this tax calculator to estimate how much taxes you have to pay.
Rent a Flat
There's several statistics about this. For instance have a look at this graph. Here's another graph which shows the average price of 110sqm flat in different parts of Switzerland. As you can see, both cities are quite expensive, but if you are okay with commuting, you can get cheaper flats in towns a bit further away.
Support for families
You will get a certain amount of money for each of your children every month. (Familienzulagen). Those are in the canton of Vaud (Lausanne) 200CHF up to age 16 and 250CHF up to age 25 if your child is studying. From the third child onwards, this is 370CHF and 420CHF. For the canton of Zurich this is 200CHF up to age 12 and 250CHF for older children. So this will only make a difference if you have more than 2 kids. Note that some of the agglomeration towns of Zurich are actually in different cantons. Here's a list with an overview of how much you get in each canton.
Schooling is a cantonal matter in Switzerland, so this might vary between the two places. I'm not very familiar with either system, but have a look here for the system in Lausanne and here for the system in Zurich. Public schools will be free at least for the compulsory schools in either place and are generally of good quality. Note that the schools around Lausanne will be mostly in French and those in Zurich in German. Although there might be one or two French schools in Zurich and vice-versa.
If you don't speak French or German, I would not underestimate the language barrier. While you might be able to find a job where you don't need to speak either language, it will be a different matter for everyday life. While a lot of people do speak basic English, you will come across people who don't quite often as English is usually only taught as a third language and especially older people won't have had any form of formal English education. From my experience, the French speaking part of Switzerland fares worse here than the German part of Switzerland. (As is also confirmed by this study).
I can't really say much about child care or buying a house. As for transport, not to worry, both cities have an excellent transport system. If you are looking for a place to live and want to find out how long it takes you to the city, you can check connections on the website of the federal railways.