The best thing to do is to read the whole sentence first and then try to understand it, which requires some reading practice so you don't get caught up in particles. I don't know if I can adequately explain how I do it, since it's mostly just intuition.
The first step of the process is probably to identify the core bits of a clause anywhere you can in the sentence, which are the subject and predicate. The subject is often left out in Japanese so it's important to figure out what it is as soon as you realize it's being implied. The predicate is the action; something like だ or any verb. て form, 連用形, the copula で, etc can separate actions distinctly in the same sentence, learn to notice that.
Commas have a ton of uses so don't be too quick to make presumptions as soon as you see one. Forget English comma usage. Listening to spoken Japanese helps because you get a better feeling for how a long sentence flows through intonations and pauses.
After you understand the overall structure of a sentence you can look closer at each subject and object and figure out how it's being modified by everything before it, and by the particle after it, if applicable. Then you can take the adverbs and the verbs and construct a meaning for the full sentence.