I moved to France as a young adult and managed to get a pretty good accent. And a trick that helped was something my (non-French) mother taught me as a child, cuz her French teacher had taught it to her and it was fun.
To make the French "u", which is one of the harder sounds for English-speakers, do this:
- make your mouth round as if to say "o"
- but say "e" instead (English e, not the short French 'eh')
Practice doing that several times a day, by itself, then in words with u in them (lu, du, culture, purée, mur, etc)
As for the other vowels, they are very short, simple sounds (not the long dragged-out diphthongs of English)
Also, the rhythm is very flat - don't go up and down all the time - there's very little tonic accent. They might go up a little at the end, especially in questions.
French is one of the hardest languages as far as the muscular gymnastics it takes to pronounce go, so you are right to want to tackle that aspect.
For more help, try these sites:
list thanx to thaliaba... in another Q/R