The standard system of units (called the SI units) actually includes the kilogram as the standard unit of mass. This standard unit system is often called the "meter-kilogram-second" or "mks" system.
You do not specifically need to use either g or kg in any specific problem; you just have to ensure that your units are consistent throughout the problem. If all of the quantities in a given problem are in kilograms, it's probably easier to use kilograms. If all the quantities in a given problem are in grams, it's probably easier to use grams. If some of the quantities in a given problem are in grams, and others in kilograms, you will need to ensure they are all in the same units (either g or kg, whichever you prefer), before continuing.
Acceleration and velocity are not universally defined in terms of "up" or "down." While it's conventional to use positive values for the "up" direction, and negative values for the "down" direction, problems can be worded in the opposite sense. A well-written problem should include information about which direction is represented with which sign, like the following:
a rocket launched upwards with an acceleration of +5 m/s2...
- Warren