What kind of calendar and clock system will we use on Mars?

Calendars and clocks are about what they are used for.  Our calendar and clock on Earth developed to support agriculture.  It would only really make a lot of sense for people on Mars to utilize a calendar/clock based on the Martian day/year if they had uses that required it.  Agriculture dependent on the sun and seasons isn't likely to be a concern without terraforming.  If sunset/sunrise and seasons on Mars don't significantly impact the colonists' lives, then there is no need to synchronize clock/calendars to them. At first, the purpose of a clock/calendar will largely be dictated by their interfaces with Earth.  If they are going to communicate to a control center and family/friends on Earth, staying connected to Earth's clock/calendar makes sense. I can't imagine why a culture completely acclimated to the 60/60/24 system would ever want to upend their lives and switch to a metric base 10 governed system.  It offers no advantages.

Some thoughts: It makes a difference if people on mars are thought to be permanent colonists, most of them born on Mars, or if they are supposed to be there only temporarily, maybe for a couple of years. In the "temporary" case, I don't suggest to try getting rid of the 12-60 mess, but rather try to keep continuity with earth. A Mars day is just half an hour longer than an earth day. So it would make sense to define a Mars hour to be a 24th of a Mars day, and the "hour experience" would be just about the same as the experience of an hour on earth. A mars year is about two earth years, so Mars and Earth are in the same ball park here as well. The division of a Mars years into Mars Months could be done arbitrarily. Pick a number not too big so the mind can still deal well with them - so I would not advise 24 Mars months of length about an Earth month; rather a smaller number like 15 Mars months, each of which has more days than an Earth month. Something that aligns well with Mars seasons might be helpful, so a Mars month typically has characteristic weather. I don't know much about the climate on Mars, though, so I cannot go into more detail here. The calendar epoch could be the date of the first man on Mars.

I agree, fractions of a Martian day would make sense.   Maybe metric from the Martian day, Decimdays as about 2.4 hours, Centimdays, .24 or about 15 minutes.    And multiples of this day, Decamdays, 10s, and Hectomdays, 100s.   And say "northwarm" and "southwarm" rather than cause confusion with summer and winter.  It would anyway be a single Martian society, and likely to be spread across the whole planet.  Not like Earth, where only a relatively small number live in the southern temperate zone where the seasons are reverse.

well I can't really see how the actual measurement of time would be different, the mechanics should be the same, the overall length of time would have too be taken into consideration, I'm sure that you've seen this article (http://www.universetoday.com/37481/days-of-the-planets/) , hope it helps

A locally relevant one...just like time zones on Earth except sols will be longer, martian years longer, etc.

Well, until people were 'bound' to Martian seasons by terraforming making outside relevant in terms of 'do I need to wear a sweater today' as distinct from 'wear an environment suit', the calendar would be the Earth calendar. After you could venture out without a suit, then the 1.8 year long Martian year (Mear) would probably get divided into 12 phodems (months), six of 55 sols (days), five of 56 sols, and one of 56 or 55 days (like February 29th, but not as often. In addition, there would be a sol inserted into the calendar on December 25th each Earth year. A Martian sol is 1 minute 33 seconds longer than a day, and would determine the daily cycle. So basically we'd have 24 mours of 60 mminutes and 60 meconds, the mecond being 1.025957 seconds. No one would actually them mours, mminutes or meconds though. It would be easy to get digital devices to use Martian time and to convert to Earth time. I would suggest time zones on Mars to be a dumb idea as it's far better for people to know sunrise is 02:00, 10:00, 23:00 or whatever where they are than deal with that.