I was asked what to start seeds in. I have cats too, and boy can they be a problem when it comes to your seedlings.
We utilize different spaces in our home, under the sink, counter tops and even the corners.
You can buy pre-made corner shelves or, build one or two. Rod's Wood Working Shop as free plans on building your own corner shelf. What he is lacking in his plans is how to turn it into a planter.
You need white paint, reflective white paint. You do not need to cover the entire shelf in white, only the underside of the shelves, this is where you will mount your lights. You can use specialty lights, like gro bulbs, or a mix of fluorescent and iridescent lighting. Keep your lighting at least 8 inches from your plants, and use a timer to give them the correct amount of light.
Use whatever wood you have on hand.
We also use a box, or closet that we built and have stashed out of the way. You can find all the information about it here at a post we call Phelan's Box
Light is super important and I do have a post all about light; What's Light Got to Do With It. A must read. . .
And I also have a post on converting an unused bathroom into a little green house, cats included in this one called Next step for the starters.
I hope those articles help you. Any questions, please feel free to ask.