How can I ensure my cosmetic injection is safe?

Cosmetic injections can be divided into 4 categories: neurotoxins (Botox), fillers (Restylane, Juvederm), collagen stimulation agents (Sculptra) or fat injections.
Safety concerns depend on the injectable that you are seeking to use.
Botox is very safe. It is derived from a poison, but, despite the recent press brouhaha, side effects are miniscule if you take into account how many of these injections are performed around the world. As long as you go to an injector (MD or RN only!) who is experienced, who uses the US FDA approved product exclusively and who performs the procedure in a medical setting with proper lighting and cleanliness, you will be safe. Doing your research into the injector is, of course up to you.
Don't be like most of the US population who puts more time into researching what TV to buy than what doctor is going to change their appearance!

Fillers and Sculptra often require more talent to administer well than Botox. The same rules apply, however. The injector should be experienced, and not just in fillers, but in the specific filler that he wants to use on you. Don't ever go for the cheap-o medical spa in the weekly free paper. You don't have to pay exhorbinant fees, but its easy to figure out what the standard price range is among physicians in your area. If you cannot get a recommendation from a friend and the internet is not helpful, pick someone in the middle of that price range. He/she should have a clean, nice office and should take the time to talk to you and listen to your concerns. Assembly line injection clinics compromise on safety and results.

Fat injection is surgery. Go to someone with excellent references and credentials (board certified in dermatology, plastic or facial plastic surgery) who does alot of fat injection in a certified surgery center. If you choose this injectable (I personally am not a big fan), do not mess around. Complications can be difiguring and dangerous.