I recently read a book (with audio material), 'Mastering the American accent'. Books like those deal with phonetics, linking, stressing words, could be a good start.

Really speaking with a native-sounding accent is hard. It's definitely useful to learn about how things work (like liaisons), but it's hard to actually use those things while speaking out loud. Personally I'd recommend just listening and talking to native speakers as much as possible, or trying to imitate what people say or how a singer pronounces lyrics as exactly as you can. Getting your mouth used to unfamiliar sounds is most important, so it becomes intuitive. Generally, I'd say vowels are the most important, and make a language sound the most authentic. They're hard to get right, but you can always ask a friend to help you with a certain sound or tell you if it sounds right. This answer is kind of vague, I'm sorry! I struggle with pronunciation in languages I'm learning too, but I hope some of this helps.