There isn't any one way to get studio work, but there are typical paths and ways to maximize your chances. Finding studio work can be separated into two categories, studio staff work and independent contract work. Let's take a look at them!
studio staff work
Staff work most likely starts with an internship (usually for school credit and unpaid) where you'll get client lunches, clean, and make coffee. But you'll also get to help set up for sessions, meet people in the industry, and get a feel for the studio business in general.
If a studio position opens up and you've been a hard worker with a great attitude, you just might land yourself a studio night staff position. The night staff is in charge of setting up the sessions, in charge of the interns, and are assistant engineers in training.
Studio assistant or "second" engineers act as a liaison to the head or "first" engineer for the particular studio room that they're working in. It's the assistant's job to know how to accomplish whatever the first engineer needs to accomplish in that particular studio room.
The assistant is usually also the "tape op", the operator of the tape machine(s) or DAW for the session. Assistant engineering is great because you're able to work with many first engineers and continue learning what it takes to be a great first engineer.
All the while, whether an intern, night guy or assistant, you should be looking for musical acts that you can be the first engineer for.
Which brings us to independent contract work...
The topic of "Finding Studio Work" really applies to the independent contract engineer in all of us. Even if you're lucky enough to have a staff engineering job, you're probably still going to be driven (financially, creatively or both) to find first engineering work.
The good news is that a lot of people want to record music - and you can find these people through social media or going out to listen to local music, even karaoke. The question isn't, "Can I find someone that wants to make a record?", it's "Can I convince this potential client to hire me, and can we make a record with the budget they have in mind?"
Convincing them to hire you will just take a confident pitch and a demo reel with examples of the quality of work you can deliver.
As for budgets...
They come in three sizes, 1) money is no object, 2) some, and 3) none. "Money is no object" is easy. Just charge them what you're worth, book a great studio, and make a great record.
"None" is a bummer, but a reality for some people. Use your best judgement here. Just make sure you feel you're getting something in return, like exposure or experience. You may love someone's music so much that you're willing to invest your time into their art, or you may need material for your demo reel.
"Some" is where most of your projects will fall. This is the area where you need to be honest with yourself and your client. It's also the area where you may be able to edge out your competition by being prepared.
Before approaching your first client, visit as many local studios as possible and get rates and gear/mic lists for all of their rooms. This way, you'll be able to quickly put together a budget for their project with some of the other following considerations. You'll have to consider the number of songs they hope to complete, how long it will take (based on their talent level and your resources), and your fee. Your fee could be hourly, by the song, or by the project.
Almost every engineer in this day and age has to have a recording situation at home. In these cases, I would urge your client to find the budget to book enough time at a studio big enough to do the basic tracks (at least drums) for their project. Doing overdubs and mixing in a DAW at home can also be a selling point. It has its obvious downfalls but allows you to still collect your fee in situations where they can't afford a studio.