However, I am trying to get a remote job now, and since I don't know anyone who works in these organizations, it is turning out to be my first real job search.
Personal network - recruiters - apply directly
You (and others) are welcome to contest this order, as I am typing it in a bit of a rush.
You should also continue to look for work on your own. I wouldn't just use a recruiter. Keep them informed about where you're applying. They should always ask before sending your résumé out but this helps them know where to focus their efforts. If you don't like working with them or lose trust in them just stop working with them. Good recruiters get a lot of business from referrals (that's how they got your business) so they want to get a deal that'll make you happy even if it doesn't make them quite as much money as another position would. In my last job hunt I ended up taking the lowest paying offer I received because it was a better fit in other ways. That cost the recruiter money in the short term but I'd recommend him to people I know if they're looking.
My personal experience so far has been pretty good. The recruiter works part-time and I can tell that this person is a genuinely good/kind person, which is why I have vested trust with this person.
As far as other recruiters go, I avoid the ones who spam jobs boards.
Can you get a job by simply applying to a job? Of course, it happens all the time. But networking always improves your odds.