For example:
put + forward = He put forward a useful proposal for change. (= proposed)
look + into = They agreed to look into all complaints. (= investigate)
work + out = Can you work out the total for me? (= calculate)
Notice that there is usually another, more 'latinate' word available. But phrasal verbs are used a lot in conversation.
How do you learn phrasal verbs?
The best way to learn phrasal verbs is the same as with all other vocabulary: see them, note them, and use them as much as possible.
With phrasal verbs, it also helps to study the contexts in which they are used. For example, a lot of phrasal verbs are used in connection with telephones:
call up, put through, cut off, hang up, speak up
Or driving:
pull over, pull out, turn on, turn round, back up, fill up
Or daily routines:
wake up, get up, wash up, put on, take off, switch on
When you read and listen to English, look out for phrasal verbs because they are everywhere in English, especially in everyday situations.
Resources
For an excellent summary of phrasal verbs, and the four different types of verbs, see this excellent page by Peter Hall.
For another overview of phrasal verbs with examples, see here.
For more exercises and a dictionary of phrasal verbs, see here.