Riding a bicycle in Calgary, you'll often notice things that make it harder than it has to be. Missing pathway connections, missing curb ramps, misplaced barriers, gravel, snow, or ice that needs clearing, missing bike racks, uneven pavement, etc. The City doesn't place these obstacles in cyclists' way on purpose: low property taxes mean little money to spend on things you don't notice unless you're on a bike. For instance, before their budget was cut back some years ago, the Roads Department used to clear Calgary's streets of gravel and debris not once, but twice (!) a year. They famously don't clear residential roads of snow, but they also don't check them for potholes or cracked pavement.
That's why it's important to report things you see to the right people. Reporting them doesn't guarantee they'll get fixed, but it makes it much more likely.
Your first step should be to file a report with the City's 311 service. There are a number of ways to do this: using a phone app, online, or by phone. If you have an Android or iPhone, the apps are the easiest and quickest way: automatically locate you, no need to enter your contact info every time. Also note that 311 operates 24 hours a day.
Things you can and should report to 311:
It's a bit tricky sometimes to find the right option in the online system. Note that everything that's on a street goes to Roads, everything that's on a pathway should go to Parks, and everything related to C-train stations and overpasses should go to Calgary Transit.
When you report something to 311, you'll get a Service Request (SR) number. You can use this number to track the status of your request. It will often quite quickly be "closed": this doesn't mean that it was fixed, just that it was sent to the appropriate business unit for action. Some things usually get taken care of quite quickly (potholes), some things take awhile (curb ramps, bike racks). If it isn't fixed after a while, you can then call 311 to inquire what the City's planning to do.
If a follow-up with 311 doesn't produce results, you can contact your councillor for assistance. They can sometimes speed things up. Include the original SR number. You can also mention it in the forums so other cyclists and Bike Calgary become aware of it. If you are on Twitter, you can also tweet at @yyctransport (for Roads), @CalgaryParks (for pathways), @calgarytransit (for LRT stations). Include the #yycbike hashtag.
If there is a recurring traffic safety issue, e.g., motorists speeding consistently or driving in a bike lane, file a Traffic Service Request with the Calgary Police Service.
If there are cars illegally parked in a bike lane, you can report them to the Calgary Parking Authority for enforcement.
Some problems may be harder to tackle. For instance, lots of things aren't under the City's jurisdiction. This includes land owned by CP Rail (e.g., the Bow pathway along the Alyth railyard), the Airport Authority, the Calgary Municipal Land Corporation (East Village and Fort Calgary), land owned by the province (Fish Creek Park, the pathway along the Western Irrigation District canal), or privately owned buildings. 311 requests will sometimes be closed without being acted upon if the City deems itself not responsible. In some cases there are legal or cost issues. But do not despair: even a private citizen can bring about change in these cases. Be persistent.
If/when things get fixed, be gracious: send a thank-you note, or an appreciative tweet.
Remember to always be polite and to be patient. 311 requests are supposed to be addressed within seven business days, but many issues cannot be resolved quickly. For instance, bike racks and curb ramps will only be installed during construction season and sometimes require permission from property owners, and potholes are only repaired when it's warm and dry. Parked cars in bike lanes used to be a significant problem because it wasn't clear that it was actually illegal to park in a bike lane. Repeated complaints by people like you eventually resulted in a bylaw change. A section of the Bow River pathway used to not be snow cleared in the winter because it ran through land owned by CP Rail. After volunteer-organized snow clearing bees, help from the area councillor, and media attention, it now is. Persistence pays off.