For NF, the flowchart in my signature might be helpful; just know that you don't have to read the entire Gateways crossover. Pick the NF parts only (Cold Wars and the New Frontier section of What Lay Beyond) and that works just fine. It's also totally fine to skip the comics; last I checked, they were a bit hard to find these days.
As far as Full Circle goes, the Voyager books that begin with Full Circle are incredible TrekLit, some of the best. They do refer to earlier things, but I think the best way to approach that is:
1) Just read Destiny. It's great, it's designed to work for new readers, and I've given the trilogy to several people who've never read any TrekLit before and they've all loved it. (It's not a bad idea to read Before Dishonor before Destiny too, but don't be too turned off if you don't like Before Dishonor; it's a pretty divisive novel, and not representative of the quality of Destiny or the Voyager books later. I usually just tell my friends to read Destiny and I haven't gotten any complaints!)
2) Start reading Full Circle and continue from there. Be aware that when you read Full Circle, you will read some references to earlier Voyager novels that you haven't read, but also be aware that those Voyager novels aren't very good and most of what's going on in Full Circle is clearing the narrative board so that Kirsten Beyer can set her own stories in motion. If a few things are a little vague, don't worry about it; those things will cease to be important by the end of the first novel, and after that you can just read the Voyager novels in order without worrying about any other references, and they're so good. A bit of a completist myself, I understand how this sounds a bit awkward, but I've also recommended this strategy to several of those people who liked Destiny and wondered what to read next, and they've all agreed - Full Circle is occasionally an awkward start, but by the end of the book they're completely on board, and the rest of the Voyager novels from there are more than good enough to justify a little patience at the beginning.