A lot of people ask me how they could become instructional designers. Here's some advice from my perspective, followed by links for other perspectives. All of this applies to instructional design in the business world, not academia or K-12 education. It's probably most applicable to people in the US.
Get experience in your current job
The first step can be to get more instructional design experience at your current job, if possible. For example, if you're offered the technical writing part of a project, you might ask about other training and support materials that the project requires and what the larger business need for the project is, and suggest that you could design those other materials to meet that need.
Another approach is to simply create what the organization needs, if you have the time. That's what I did in the 1980s. My job was a combination of tech support, technical writing, and training, so I was in a good position to identify ways to improve the organization's performance. By volunteering to improve processes and create training and support materials beyond my official job description, I created items for my portfolio and eventually transitioned into a more "pure" instructional design role.
Another approach is to politely offer to overhaul an existing course or other learning intervention that isn't working or that people complain about, even if has nothing to do with your job description. That way you can show what you can do, save your colleagues from suffering, and learn how to redo others' work without stepping on toes, which is a valuable skill in our field.
Build a portfolio
A lot of designers create portfolios that show examples of elearning that they've developed. Maybe they do this because it's easiest to show elearning, but they often give the impression that they see themselves as elearning developers and not as instructional designers.
If you want to do instructional design, meaning you want to solve performance problems and not just be a course producer, be sure to explain the instructional decisions you made for each sample in your portfolio. I've looked at a lot of ID portfolios and by far the most common mistake is to simply show the material without discussing the strategy behind it or linking it to any larger need in the organization.
If you currently do only design and not development, put your design ideas in the portfolio. For example, describe the performance problem and the solution that you designed, explaining your reasoning and describing the results.
Consider volunteer work
If there's no way to get ID experience at your current job, you might consider doing some volunteer work with an NGO or LINGOs. The UN online volunteers program might also be a source of projects or contacts.
Again, it would be important to design the project to support the organization's strategy and not let stakeholders treat it as just a writing and media task.
Decide: design or development, or both?
As you get more experience, identify what gives you the most satisfaction. Do you love analyzing a performance problem, figuring out a solution to it, and outlining a training program that you know will be effective? Or do you love to create the media for content that already exists, making it more interesting and interactive?
Most jobs that I hear about seem to require both sets of skills, which means neither gets the time it deserves. If you like both the instructional design and the media creation, there are jobs out there that will give you the chance to do both, keeping in mind that you probably won't have time to do your best work. The number one complaint I hear from solo instructional designers is that they're not given enough time for strategic instructional design and are expected to just put existing information online.
If you prefer strategy and design over production, look for a position in an organization that's big enough to have a separate production team or that outsources the production. Also make sure that your possible employer will view you as a performance consultant included in training strategy and not just as someone who converts information into a course.
Learn the theory
If you aren't doing it already, I'd recommend that you also do some independent reading to get the vocabulary and basic theoretical background that's expected of IDs. Cammy Bean lists a lot of recommended books and sites that traditional IDs use. I got the most help from Michael Allen's Guide to E-Learning, Ruth Clark's research-based books, David Merrill's First Principles of Instruction, and guidelines for performance consulting (Performance Consulting: A Practical Guide for HR and Learning Professionals by Robinson & Robinson is one book).
Since many people in our field like to discuss theory, it's good to be able to talk about Bloom's taxonomy, Gagne, constructivism, etc., much of which is described for free online. However, I think it's equally important to remember that most of it is theory, and that experiments that support a theory were likely done on students learning facts in school, not adults learning to do something in the business world.
Consider a degree
Compared to other instructional designers, I seem to be less enthusiastic about degrees. I worked with several recent graduates of ID degree programs and usually found that they learned a lot of theory but had little knowledge of business needs and no experience applying the theory to real-world business situations.
If you think a degree will help your career, I'd encourage you to be skeptical of programs that lump educational and corporate instructional design together. Based on my experience in both sectors, they have very different goals and require different approaches. Applying educational approaches to business problems creates the irrelevant information dumps that so many people complain about. The greatest weakness I see in corporate instructional designers today is a failure to consider the needs of the business and therefore the need for behavior change, and I blame the academic "knowledge transfer" focus for that.
So if you want a degree, I'd strongly recommend that you consider programs that give you real-world business projects in addition to the theory. The program should also teach you about the metrics used to measure business performance. You shouldn't spend thousands of dollars on a corporate ID degree and graduate without knowing what a "key performance indicator" is.
I'd also encourage you to look for any pseudoscience that the instructor is presenting as fact. Does the program say that you should accommodate learning styles in your design? Do they apply fictional percentages to theoretical models to make them look science-y? Do they treat theory as proven fact? If so, go elsewhere.
For example, the most recent program I checked teaches that learning styles should be accommodated, which puts it many years behind current research. As L&D departments become more serious about using valid, proven techniques, graduates of such programs could find their degrees actually getting in their way.
Finally, I'd be leery of a graduate degree program that teaches you how to use certain tools. I don't think software training belongs in a graduate program, especially when people in the real world learn the software on their own for free. Your education should equip you to diagnose performance problems and help people change what they do; you shouldn't pay thousands of dollars to have an instructor tell you which buttons to click unless you're in a certificate program specifically designed to teach you a tool.
I haven't looked for an ID job for a long time, so I don't know how many employers require degrees now. However, I can say that when I helped evaluate instructional design applicants, I didn't see a strong correlation between having a degree and having a portfolio that shows effective design decisions.
Here's a very thorough guide including the theory you should know, from Devlin Peck.
This article from the Elearning Guild lists several degree programs and includes tips. It's written from the conventional perspective that instructional design is about knowledge transfer and is basically the same for both the education and corporate sectors; I disagree.
Christy Tucker has a series of posts on Getting Into Instructional Design
10 Qualities of the Ideal Instructional Designer; includes discussion in the comments
The Accidental Instructional Designer; includes lots of links to more information
Do You Need an Instructional Design Degree? with many comments
Instructional design and elearning programs: a long list of US programs. Consider asking for opinions about specific programs in one of the LinkedIn groups on elearning.