Here's some advice I'll throw against the wall...
I second the notion of tailoring your pitch to the publication - absolutely, in all cases. You have to give them a reason to run it. Local hook, anniversary, similar event taking place simultaneously with publication date, etc. And know the lead time - sometimes it is three months or more for a monthly.
In regard to writing on spec. I was asked to do so once, for the very first story I ever tried to sell. I did, and if I hadn't I probably would be doing something else, but I never did again. If you already have substantial clips, don't ever.
Nearly everything I learned about how to get started came from a book that was at least twenty years old called, I kid you not, "How to be a Free Lance Writer." I had no connections, no clips, nothing but an idea. But apart from updates in technology - email vs. the post office - the same basic advice in that book still holds true in regard to putting together queries, targeting publications, etc. One of the great lessons from that book was the advice to learn how to sell the same basic story again and again. At first I couldn't see how that would work, but since then I've learned to re-frame an original idea based on audience, market and form. I've done that a number of times, selling the same basic story, without repeating myself too much, looking at it from a different angle each time, or even taking the same approach of a successful story and reapplying that to a different topic, event or subject. Every time it was a bit of a different take, aimed toward national, regional or local audience, in genre magazines, general interest and books, as columns, features or profiles. I always included word counts in my pitches, then hit them as hard as a deadline. Speaking of which, when given a deadline, I try to beat it if I can have a finished product that doesn't appear rushed. Many editors appreciate this, and remember it. For publications that fact checked, I virtually footnoted for them to make the process easy. It is easy enough to get rejected - don't ever give an editor any additional ammunition to do so.
It takes time to build up, and time to go full-time. Rejections are part of the process. I've had hundreds and still get them. It was seven years after I sold my first story before I gave up the day job, and even then, I made less than $10,000 that first year, fourteen years ago.
Also, don't immediately reject small, low-paying gigs. Several of the smallest jobs I've done have, over time, turned out to be huge. I once accepted a work for hire job a friend rejected for a couple of thousand dollars. It led directly to a series of assignments that over a decade have netted me low six figures, and led to significant other work. Editors at small publications sometimes become editors with big ones. I still write some things for free, and some for less than $100.00, just for exposure. That has paid off over and over again, as your work reaches some people it may not otherwise, and in that little italicized section at the bottom, or in the contributor notes, you get to promote yourself and your work. It's advertising for yourself and your work.
One last thing. Get your own web site. The father of one of my daughter's friends offered to set one up for me, for free, several years ago. He wanted to use it as an example to get other work. I said sure never thinking it would get me work, but over the past five years I've probably made $25,000 or so on assignments I received from people who discovered my on the web. Make sure it includes a list of publishing credits and some sample work
The biggest difference between people who want to write and don't and the people that do, is that some quit, and others don't. In freelance work, this is even more pronounced.