Indoor gardening is a bit more difficult then the regular outdoor gardening, there are several factors but in most cases it boils down to: Temperature, Humidity, Light, The air in your apartment.
You didn't mention exactly where your dying plants are being placed. Do you have a fan within the apartment? Plants need air whether it is natural or even if it is coming from a fan. What type of lighting is inside the apartment? Are you only using natural light? These are important factors to aide the growth of the plant as well.
How is the temperature within the apartment? If you notice a lot of temp ups / downs in the area you are gardening than it is probably worth moving the garden or choosing another spot. With apartments that becomes all the more difficult.
Your plant also probably came from some nice greenhouse, an english gardens, a ray weigands, etc etc...They have plenty of light, they also have plenty of means of water and the temperature well most of this is all outside, so they have the advantage of using mother nature. Your plant needs to adjust to these conditions. You should try to begin to place your plants in a very lit area with as much natural wind / fan as possible, these plants need to breath. If your plants are not conditioned for this type of environment you run the risk that they give up and die.
When potting the plant ensure the actual pot is cleaned up to remove any unnecessary plant diseases from prior plants. Look on the bottom of the actual pot, are you getting drainage? If not drill a hole in the bottom of the pot if you can, if not toss that pot out and get you a pot that has a good amount of drainage. Overwatering plants will destroy them, and without nature (wind, temperature, etc) the water sitting there takes all the longer to dry out.
When it comes to veggies Tomatoes need a ton of sunlight and humidity is great for them as well. Herbs are probably the simpliest to get going but I'm not sure if this is what you are after. Cucumbers and squash don't need as much sunlight as tomatoes.