First off apartments arent the best place to grow, I know, it can be a bitch.
Your gonna need a source of light, CFL's (compact flourescent light) are cheap, easy, low heat, and fall into the catagory of simplest tools. You'll probably want to invest into a good HID (high intensity discharge) lighting system later on, but for now go easy.
Air is a must, you need fresh CO2 rich air for your plants, which can be a bitch in a closet. Depending on how many plants you have a fan, and an open window will work.
Soil, hydro, aero, DWC, it's your choice, I would reccomend soil to start.
Fertilizer, Organic or chemical, take your pick. I would reccomend organic, but do your thing man.
Knowlege is the best tool of them all, there is no replacement for personal experience. But reading is the key to success, you've come to the right place. Go check the "GROW FAQ'S" section. But personally I like these grow faq's better (no offence cannabis.com, you know I love ya) The Cannabis Cultivation GrowFAQ - The Internet Hub for Marijuana Cultivation - powered by phpMyFAQ 1.6.5 , also go read around the posts in the forums, the little bits and pieces you figure out while lurking in other peoples conversations add up.
Good luck getting started.
peace