The usual advice applies: unplug appliances when not in use; use CFL light bulbs; set your thermostat low in winter and high in summer, and adjust it at night for when you're snug in your bed. You can even keep throws and blankets around living areas in the winter; they'll allow you to save on heating and have the added benefit of giving you an excuse to cuddle with a loved one.
Turning off the TV, computer and video game player can save a lot of energy, too. Consider inviting friends over to play board games, charades, or cards instead of surfing the internet or flipping channels. A regularly-scheduled candlelit evening can be a good way to foster conversation and add a little romance while decreasing your energy bill.
Transportation and food are two of the biggest emitters of greenhouse gases. Consider getting a bike for transportation if you do not already have one; if space is a factor, there are pulley systems to raise your bike to the ceiling for storage, or you can buy a cheap hook at a hardware store to hang it from the wall (be sure you get that hook into a stud!). A used road bike can usually be found for very cheap, and they make reliable commuters. Many people have quality bicycles from the 70s and 80s in their garages, and they may be willing to give it to you, so let others know you're in the market. Take any bike to a reputable bike shop and let them advise you. If you find a seller online (craigslist.org is a good place), arrange to meet near a bike shop so you can have them look it over before you make your counter-offer, or take along a knowledgeable friend.
As for food: while it may make your home energy bill higher, cooking at home will lower your carbon footprint dramatically and can improve your health as well; luckily, the most environmentally-friendly foods are often among the cheapest and healthiest if you eat low on the food chain (lots of whole grains and vegetables -- go easy on the meat and dairy). Consider getting a pressure cooker from a thrift store to quickly cook your own beans. If you have a balcony or a south-facing window, grow an herb garden. You can easily grow common herbs like parsley and basil in small pots on the windowsill or in hanging baskets. When purchasing groceries, look to local growers: many cities have nearby farms with Community Shared Agriculture (CSA) programs which provide a share of the crop for an up-front investment. Your city may have community gardens where you can rent a small plot of your own; if you're really ambitious, you can start a community garden of your own in a vacant lot nearby. Look around for people with fruit trees in their yard. Many people are overwhelmed in the fall when their trees start producing, so ask if you can harvest some for your own use.
Furnishing is another area that you can save a great deal of money and be good to the planet. Many people throw out perfectly-good chairs, dressers, and even appliances, and by salvaging these items you spare the earth the energy used in producing and shipping new goods. Figure out when your city picks up large curbside items and get on your bike the day before to patrol the neighborhood, and keep an eye out while driving around. Near the dumpsters of apartments and university housing can be a great source as well; make regular scouting trips at the end of semesters (the best time is before summer break). Learning some basic repair and upholstery can give you a stylish home for cheap; my wife and I found several vintage chairs in a church dumpster and replaced the vinyl seat covering with designer fabric, and we get compliments on them all the time.
When it comes time to throw things out, re-use and recycle as much as possible. If you choose to buy frequently-consumed packaged goods (e.g. peanut butter) in glass containers, you can clean them out and use them for storage. Unusual beans look great displayed on a shelf in the kitchen. Look for other ways to reuse packaging you might otherwise throw out, and post any large items (or things like packing peanuts) to your local chapter of FreeCycle or on websites like craigslist.org.
You can even compost in an apartment. A worm bin is easy to make and allows you to compost food scraps in a small space with little to no odor using a plastic tub, red worms, and newspapers for bedding. See the link under "Web References" below for more detailed instructions. The resulting compost can be used for your window garden or dumped into the apartment complex's landscaping. If you throw your scraps in the bin instead of down the garbage disposal, you'll save electricity here, too.
There are many things you can do to live in a "green" way without having to sink money into a rented property. Use your imagination, and you'll come up with more on your own!
You can start with the obvious: take steps towards the efficiency of your utilities at hand. Since you do not own the furnace and AC unit you are limited to keeping the air filter updated keeping the thermostat down to conserve energy there. Of course there is also the changing of the light bulbs. I've been making it a habit to turn down the thermostat when I leave for the day.
Talk to the owner to see what he/she is willing to do, i.e., new windows, maintenance appointments for the furnace, weather sealing and beefing up the insulation.
If you do not own the property you are living in there's very little I would advise in the way of modifying the structure to make it greener. You will leave one day to move to something else and the money and efforts you put into where you are leaving now will belong to the owner and you've lost in that respect. The condition of the structure is the responsibility of the owner and if he/she does not want to take the property green, there's very little you can do to compel these steps.
Now, if you are thinking about purchasing in the future and you want a property that is green, there are so many websites that offer ideas and advice on greening up your home. The thing is that you could spend so much time looking for a green home, that it might be a better idea to look for a loan that will include provisions to update the home you are buying. It's a headache and getting the equity out of your newly updated home over the next few years is likely not going to happen, but in the long run (ten years or so) it will pay off.