I have seen a lot of websites and some books that explain what phrasal verbs are and then give mostly examples. But is there a general way to learn how to form a phrasal verb and especially to get the real meaning of the phrasal verb after adding, up / with… etc?
You won't find an easy way. You'll have to learn them as a part of a phrase, or as vocabulary, mostly.
However, here's a list of the most used particles to construct phrasal verbs.
back = repeating or looking into the past.
Could you play back the telephone message, please?
The tennis club dates back to the 19th century.
down = record in writing or reducing.
Could you write these dates down?
House prices have finally gone down.
off = departing or ending.
He quickly said goodbye and then ran off to catch the bus.
The two countries have cut off diplomatic relations with each other.
on = continuing or attacking.
She spent the whole time going on about her new partner.
Those bullies are always picking on Dave.
out = disappearing or solving, searching.
The forest fire finally died out after two days.
Can you work out the answer to this math question?
over = visiting or considering, examining.
We popped over to my mum's but she was out.
Let's go over the report before the meeting.
up = approach or improve.
He always creeps up on me. He just appears without making a noise.
I want to brush up on my computer skills this year.
I am discovering that deeper, more profound answers require one to appeal to Linguistics. So search for or Google "semantics of phrasal verbs", to read some of the linguistics research papers, but which may presuppose knowledge of the branch of Linguistics called 'Semantics'; so maybe you can read an Introduction or two to Semantics first?
His 1971 book The Phrasal Verb in English, heretofore a subject of concern primarily to teachers of English as a foreign language, brought the need for a scientific treatment of phrasal verbs to the attention of many linguists.