No, a sediment filter will not remove calcium -- the only thing that will is a water softener. These use a media that only needs to be replaced every 10 years, and a brine tank with salt, which you have to refill periodically depending on water usage and hardness levels: anywhere from a couple weeks to a couple months. There's really no other regular maintenance that normally needs to be done on a softener.
Softeners substitute hardness (calcium carbonate) as well as some heavy metals, lead, manganese and low levels of iron with sodium ions (salt). The drawback to this is that increases the sodium levels in the water, which can be a concern for people on low-sodium diets. The amount of sodium added is entirely dependent on the amount of hardness removed.
For this reason, people often run a bypass to the kitchen cold water tap so it gets hard water, as well as outside taps used for watering the lawn and garden.
Before you purchase, you should test your water for hardness. The easiest and cheapest way is to buy a hardness test strip kit: you dip it in the water, match the colors and it tells you the hardness level. This will help you sizing the softener to buy. It's also good to periodically check so you can ensure the softener is set correctly. You're on city water, but depending on where they get the water from the hardness could change seasonally.