your reputation is your main asset.
+1 for this, definitely. Most of my customers are companies where someone has seen my work before and recommended me.
To the OP: you won't be able to be too selective, unless you have such a great reputation and such a wide network of industry contacts that there's way more work available to you than there are hours in the day (in which case, please give me a call!!)
Every customer will expect that you already know their product area well. You'll need to be able to quickly identify what their area actually is (because they'll never tell you, it's just assumed that you're already completely familiar with it and that you rarely do anything else), and you'll need to be able to appear confident while at the same time making a mental list of things to read up on.
This is where experience really counts. If you've actually worked on, say, industrial automation products before, then you'll be in a good position when someone asks you to design parts for a robot. But if all you've ever done before is radio networks, you'll be stuck.
Tip: it's better for all concerned if you turn down a job you know you can't do well. Don't bullsh*t.
That said, every project will include something which you've not done before, so be prepared to learn. This is true whether you're contracting or just doing a new project for a regular employer, and nobody will expect you to have built something that exactly matches their product before.
At 24 I'd only just graduated, and although I was keen enough, I had an awful lot of commercial and industrial experience still to gain. I really don't think I'd have been able to pick up a project and immediately debug, progress or redesign it unassisted.
Building stuff on your own is a great idea - pick a project area that's interesting to you, and make sure it's something you can't already do. For example, I've done two in the last few years. One was a hi-fi DAC, which taught me a lot about low noise power supplies, and the other was a motorcycle accessory for which I had to learn all about how to use a PIC and program it in C. Neither product is commercially available in its own right, but I've drawn on both for inspiration when customer jobs have had similar requirements.
It's also handy to have something you can show a prospective customer without breaking NDAs or confidentiality agreements. Design your own PCBs and get them made properly if you think you might ever show them to prospective customers. Nothing says 'amateur' like an Arduino.