I am going to try this too this year. I start seeds under lights every spring, and I keep a light on in winter to overwinter a few flowering plants, but I've never tried to grow food completely under them before.
My plan is to grow a little spinach, romaine, and mesclun mix in those rectangular plastic containers you can buy salad from the store in with holes punched into the bottom, to be harvested as baby leaves, mostly, rather than full grown plants. I also want to grow elephant kale in six inch or one gallon (I'll try both to see if it does better in larger pots or if the smaller are good enough) mostly because it's difficult to find in the store. I may try some beets and/or carrots, but these would need taller containers. I like prefer yellow/golden beets though just because I don't have to worry about the kids staining their clothes with them the way they do with red, and those aren't as common in stores.
I leave my lights on eighteen hours a day when starting seeds, but I think I'm going to try for just sixteen over winter to save a little power. You need the light fairly close to the top of the plant, so when growing short and tall plants under the same light, I use piles of old textbooks (with a plastic tray on top) to raise the shorter plants up to the lights. I also have tinfoil on the sides (my lights are set up on utility shelves, so I tape it to the side of the shelf above and let it hang down) to help reflect more light back onto the plants. My lights are in my basement, so they are the only source of light.
Peppers need a ton of light and heat. They are hard enough just to keep alive overwinter under artificial light; I don't think you will have any luck getting them to flower/set fruit unless you have expensive HID lights and are prepared to spend a fortune on energy bills.
You might have better luck with tomatoes - mine have always done very well started under light, much better than pepper starts, but I've never tried to keep them inside long enough to flower. You probably want to try with a tumbling variety or other intended for pots to be able to keep more of the plant under the light - some of my tomato plants outside are currently taller than I am, and indoors the lower leaves would not be getting any meaningful light from overhead lights. When growing tomatoes inside, because there is no wind you want to either brush your hand over the top of the plants frequently or run a fan a few minutes a day from different directions to make the plants grow stronger stems. Also, tomatoes are self fertile so you don't need to worry about insects to pollinate, but again when the plant is in flower you'd want to gently shake it a bit to help the pollen move around within the flower to where it needs to be.
Flourescent lights, even full spectrum ones or grow lights meant for plants, don't produce nearly as much light as the sun. If you have them near windows so get natural light as well, that would be better, but even then the light won't be as intense in winter as it is in summer. It would be fun to test it out and see what you can get to grow in them, but you might wan to start with just a plant or two of each before you spend money on more potting soil and take up space under lights for a bunch of plants that may not do well. For the most part, you are probably better off looking at lists of vegetables that will tolerate a bit of shade and going from those.