Hi - I was a primary Teaching Assistant for twenty years. Sometimes children from a non-English background can make BETTER progress because parents and child are prepared to put in more effort than an English child of similar age.
Yes, reading as widely as possible is the first thing. As someone said a THESAURUS is useful, but a book, not just on the computer. You can get children's versions, and it shows you lots of words that you can use instead of the simple word. So, suppose you have the word HOUSE: other words might be home, cottage, bungalow, flats, mansion, palace, dwelling, residence, abode, hut, shack, pad (slang word). etc. Depending on where the word is to be used, some words are suitable, others are not.
You can also get a children's 'Rhyming Dictionary', so you look at one word, and it shows you ones that rhyme with it; that might help you find words you wouldn't normally come across.
You can also get free on-line: www.rhymezone.com/r/rhyme.cgi?Word=house&typeofrhyme=syn&org1=syl&org2=sl&org3=y
In the MN 'Book Reviews' section, under children's educational books and courses I did a review of a Phonics Spelling Dictionary, which will help a lot with English spelling (which can be difficult for some people.)