First..let me say this. I am a laser engraver and I have to have all cut lines as vectors. The machine I have has to have specific line weights to recognize as vector lines to be cut. I use both Corel Draw and AI CC , as well as Photoshop in my work.
In order to convert your Photoshop or bitmap content into usable, editable vectors, one way would to be use Live Trace. Once you run Live Trace you have to Expand the file. You can then manipulate the anchors with the Direct Select tool. You may want to Simplify the design to reduce some of the anchor points. You can then add whatever stroke you want.
You can also do a manual trace with the pen tool. It works just like the one in Photoshop. You can assign strokes and fills to the lines just as before. This with be what I would do to outline the one you are displaying. That would just be a simple line around the outside for cutting it out.
I must say though, I find AI very cumbersome for my actual cutting process. I love the tools in AI and it has some things that are fantastic for my purposes, i.e. the Blob Brush for one. I do my designs in AI, convert them to PDFs and bring them into Corel where I am more comfortable controlling my placements and sending functions to the laser. I have some neat macros that I have acquired over the years that allow me the luxury of some functions I can't do in AI, such as determining the total length of a curve. With that information, I can tell within a few seconds of how long it will take to cut a file.
Oh, and by the way, when I got AI I was about as lost as I could be. I found an online course, produced by Infinite Skills that helped me tremendously. You can join that thing for $25.00 and only stay a month if you like, but the course is pretty good and you get access to all the content for the money. They have complete courses on many computer related software programs and photography, so I have kept my membership to learn more stuff. Check it out. Just Google it.