I am thinking of putting a 120mm stem on my Safari (currently 100mm) to get the handle bars a little higher. How will this effect steering? Will this push the handle bars closer our farther away from me?

I don't know, I can't see it from here.

haha
you'll have to give the rise... Two stems of the same length can give different effective reaches depending on their rise. A higher rise stem will decrease your reach. If what you want is higher bars, you might decide to play with your headset spacers, stem length/ reach, etc.

Thanks for chiming in FbB. sa2405, modern stems are usually described with all of the following:
1) fork steer tube diameter
2) length
3) rise, in degrees from the steering axis
4) handle bar diameter
Obviously, length and rise are all you have to work with here. Depending on the rise, you could get the bars closer to you and higher, further away and the same height, or further away and lower with a 120 mm stem. I would go to a bike shop and see if they have something cheap or used to try out, swap, borrow, or purchase outright. I don't think 20 mm will effect the steering too much, but you might be surprised and are smart to ask that question. Back in the pre mountain bike era any thing over 120 mm was somewhat rare, and considered a detriment to handling.