Study the particulars of the genre by renting a variety of Sci-Fi films that have either been done well or have been laughably awful. The Internet Movie Database (Resources) will get you started with a list of the very best and the very worst. As you watch these films, pay attention to the structure and pacing of the story, the incremental escalation of suspense, the use of lighting to create mood, and the incorporation of sound effects and music. Visit "The Movie Cliches List" (Resources) and take note of how many of the incidents identified in the links to aliens, asteroids, Independence Day, and spaceships occur in the films you study.
Decide what kind of Sci-Fi film short you want to make. Under the broad umbrella of Sci-Fi, you not only have the genres of comedy, drama, mystery, romance, etc. but you also have a template of popular themes that lend themselves well to futuristic/alternative universe tableaus. Examples: (1) mad scientists and lab experiments that go horribly wrong; (2) alien invasions, (3) genetic mutations (including clones); (4) raising the dead (which runs the gamut from prehistoric creatures to zombies); (5) apocalyptic disasters and rebel societies, (6) time travel that takes a wrong turn, and (7) evil computers and robotic mayhem. Another option is to do something completely silly that spoofs existing Sci-Fi films. One of the funniest around is Grocery Store Wars which can be found at the Store Wars website.
Write your script. At 15 minutes, it's going to be 15 pages long and should have a definitive beginning, middle and end. Example: In Act 1, the hero's girlfriend is abducted by Martians; In Act 2, the hero's family and dog get abducted, too; In Act 3, the hero infiltrates the Martian camp and is nearly killed by the Martians but succeeds in vaporizng all of them, rescuing everyone, and saving Earth.
Identify your various shooting locations. (For a film as short as this one, you're not going to have that many.) If, for example, the action takes place on a desolate planet, go to a beach, an overgrown backyard, or put a miniature spaceship in a sandbox and do a tight shot. If you're shooting apocalyptic scenes, go to an empty playground, underground garage, or parking lot. For mad scientist motifs, see if the local high school will let you borrow the classroom that's used to teach chemistry. For Sci-Fi's that travel back to primeval times, any dark forest should work.
Collect the various props you're going to need. Any shop that sells Halloween costumes is going to have spooky rubber masks, capes, fake ears, and makeup supplies. Craft stores carry all manner of pipecleaners, popsicle sticks, Styrofoam shapes, boxes, and fairy lights. Raid your little brother's toy box for plastic dinosaurs, Legos, rubber snakes, and plastic bugs. Check out the refrigerator for squishy fruits, weird looking vegetables, and colorful liquids that could stand in for alien "blood." A world globe that's set against a black sheet will look like Earth from outerspace. Blowdryers make convincing looking intergalactic weapons.
Decide what kind of sound effects and music would enhance your Sci-Fi movie. Although sound is usually added last after everything has been shot, it's not too early to start thinking about what you'll need and making notations on the script as to where the different sounds go. There are plenty of websites on the Internet that allow you to download sound effects for free. These include things such as sirens, noisy crowds, traffic, animal noises, screams, and machinery.
Recruit actors for your film. In addition to like-minded Sci-Fi lovers who comprise your friendship pool, you can also ask your family members to participate or put up a notice at the local theater that you're having auditions. If you don't have the resources to pay them anything, it's a professional courtesy to at least provide snacks and beverages during the shoot and provide them with a copy of the film when it's done.
Set your production schedule and make sure everyone is available. A script this short shouldn't take more than a few days to shoot and edit. Filmmaking software can be purchased through venues such as Rose Lantern and Filmmaker's Store. If you have a local access station in your community, you can go an even cheaper route by inquiring whether you can use their studio for your post production needs.