Da generally does mean "from." Dal is a contraction of da and il to make "from the."

Your question is quite challenging to answer fully but the following are perhaps the first steps towards an understanding:

First of all the contracted forms of da + article:

• da + la = dalla • da + il = dal • da + lo = dallo • da + le = dalle • da + i = dai • da + gli = dagli

Da can also mean "to" or can be used to make a word adjectival eg sala da pranzo.

Below are the contracted forms of di + article:

• di + la = della • di + il = del • di + lo = dello • di + le = delle • di + i = dei • di + gli = degli

Di usually means "of" but typically confusing of prepositions can also mean "from" or "about"

There is, in fact, a lot more to it than this but maybe get this first then check out a good grammar resource to consolidate and improve your knowledge. Hope this helps :)

I think you use ad when the next word starts with a vowel. Is that right??

Because da mostly means from. Also if you want to say "from-to" it's always "da-a" or with the article "dal/dalla/dall'-al/alla/all' "