... how could I increase the shutter speed and lower the ISO at the same time?...
Easily. The flesh exposure is influenced only by the distance and f/stop (aperture), not shutter speed. Unless your subject is far away, you can easily use a lower ISO.
Shutter speed should be somewhere between the lowest hand-holdable speed, i.e., the one that would avoid camera shake (usually the nearest reciprocal of the focal length in 35mm terms... in your case, with your zoom at, say, 105 mm, it would be 1/125 s) and the maximum sync speed of the camera (usually between 1/60 s and 1/250 s).
You are correct, I was trying to capture a lot of ambient light and use the flash at the same time.
That is not a bad idea in itself. You avoid the harsh, flesh-only light that way. My suggestion to lower ISO was meant to eliminate the ambient light, if that is what we want. If we, on the other hand, want to keep it, than ISO should be increased, of course, in order to gain a faster shutter speed.
One reason to avoid too much ambient light in combination with flesh is the different color temperature (white balance). Flesh illuminated parts would be daylight/flash balanced, typically 5500 K, while ambient light might be either yellowish (e.g., tungsten, 3200 K) or greenish (fluorescent) or some combination of both. One way to deal with the different white balance would be to use colored gels in front of the flash. Say you are shooting with tungsten as ambient light, you would then put an orange gel/filter on your flash. In case of fluorescent, green filter.
As you can see, it can get pretty complicated pretty quickly
It is getting late here, and my explanations above might not be the clearest possible, so let me try to summarize:
To avoid ambient light > use faster shutter speed, smaller f/stop and lower ISO
To include ambient light > slower shutter speed, larger f/stop, higher ISO
To balance ambient and flash color temperatures > use colored filters on flash