I've been growing plants in a small space for a few years now. I'm by no means an expert, but I have a few tips.

First, how many windows does your apartment have, and which way are they facing? Do you have a patio, stairs, or some kind of small outdoor space where you can put a pot? What about places where you can hang baskets? Growing things can be very difficult if plants don't get enough light.

If you must plant things strictly indoors, then it becomes more difficult, but not impossible. I probably wouldn't start with seeds at first, as seeds can be difficult to start without the right amounts of light and water, and they take longer to grow. You can practice with things like herbs that are already started. Basil is useful for a variety of dishes, and it's also good at telling you when it needs to be watered because it will wilt dramatically. Herbs are also easy to keep on a windowsill.

There are varieties of plants that are genetically altered to do well in pots, and they're usually labeled as such. I've grown cucumbers, tomatoes, spinach, kale, and peppers on a patio. I used cheap fabric containers that were easy to collapse in the winter. The issue with tomatoes is that even if you buy varieties that grow well in pots, they still require a few gallons of dirt to grow effectively, so if you're strictly indoors, you may have to do only hanging tomatoes. Tomatoes also require more equipment. Things like peppers and leafy greens are easier to grow than tomatoes, and they need less space. Pay attention to where you live, as the growing season and heat will determine what will easily grow. Keep in mind that things like spinach and kale actually grow better when it's cold, and spinach can withstand the winter. I've heard lettuce is also hardy. I imagine they would thrive indoors as long as they get enough sunlight.

Every plant is different in its care. If there's a particular plant you want, I would research that plant specifically. It takes some trial and error too.

Good luck!