It's a difficult problem. Luckily, modern English is much less restrictive and few people will notice the difference. It's pretty common for natural English speakers to say things like "they considered her part of the family", or "Come join us and be a part of a winning team", for example. On the other hand, with some sentences it is important.
The best method is probably going to be to just practice lots and get a feel from it. I think this case (using 'part') in particular is especially tricky, and there's no entirely strict rules that allow you to tell the difference.
The final example is perhaps the most important: qualifying 'part' (with 'important', or 'minor' or 'major') forces us to use 'a' or 'an'. For example:
"Come join us and be a key part of a winning team" requires us to have the 'a'.