the first thing you need is contacts. a lot of people tell me they don't have contacts, don't know anyone, etc, but that is almost certainly untrue. start talking to your friends, family members, and literally every one else you know or have ever known. facebook is a great way to do this. if you can find someone with an in in the publishing industry, starting your freelancing career will be a million times easier.
if you truly know nobody, and you have exhausted all your resources, i suggest getting on twitter and following some editors of magazines and publications you like/want to write for. interact with them (don't pitch them immediately) and eventually they'll start to know you/who you are. this is a way of getting to know people that's pretty easy so long as you don't come off too strong, but it's nowhere near as good as having even a very tangential in-person "in".
as for the process -- i can only speak for my type of writing, which is journalistic/longform freelance writing. i write for magazines, both online and off. for me, i try to invest as little as possible in an idea before i pitch it and have a price on it. that's just part of the hustle -- no point in spending a lot of time on something that may not even see daylight. this is different for longform pieces, like those profiles you see in rolling stone. also different for anything that revolves around a single, important interviewee, for obvious reasons. no point in pitching an interview with brad pitt if you don't already know you'll be able to talk to brad pitt.
ok. so here's what i do for my shorter pieces, which are my bread and butter. find an idea -- i like to pitch 3 - 5 ideas to an editor at a time. pitching just one idea and they might not love it, which means i might not get a callback. i want to get at least one or two callbacks for each pitch letter i send -- not only is it good for me (obviously), it also instills in the editor's mind that my emails are the ones to read. if i pitch five different ideas in different emails and my editor takes two, then that's 5 emails my editor has opened and only two they've liked. if i pitch five in one email and my editor takes two, that's one email they've opened, one they've liked.
but, of course, too many ideas and it's overwhelming for the editor because, realistically, editors have a budget for each month. if i pitch too many ideas at once, and they like a few of them, that's a much longer and more detailed email they have to send back. you want to make your editor's work easy.
once pitches are accepted, then i start doing the research. prior to pitching, i do just enough research to know (a) what i'm talking about and (b) if my idea is viable within a reasonable time period.
i know it's different for a lot of people -- you'll see a lot of things about query letters and all that. i don't think query letters are ever a great thing, if you can avoid them. they're long and they imply that you don't already have a handle on the editor's beat. i like to have a relationship with each editor i work with, even before i start pitching them.