Well... I think you can beat most of this by upping the ISO to begin with. I would try setting the ISO to 800 and see how that goes. You'll want your shutter speed to be 1/150 or higher to avoid shake and blur. You should be able to achieve this without any fancy lenses. However, you might want to get a 50mm f1.8 lens. It is a great all around lens that can work well in very low light given its 1.8 effective aperture... and it only costs around $100 used/new.

Also, the easiest way to get faster speeds is to use a fill flash... perhaps you can get a SB600 flash unit to help bump up the shutter speeds. It will help fill in the shadow spots and give you the boost needed.. use indoor and outdoor.

For lenses - I highly recommend the following:

18-70 "kit" lens - ~$200 used
17-55 f2.8 - ~ $975 used
18-200 f4.5-5.6 - ~$550 used
70-200 f2.8 VR - ~$1475 used -> amazing lens.. fast with VR.. but heavy and $$$$$
50mm f1.8 - ~$90 used -> great lens for low light situations .. and great price
60mm f2.8 macro - ~$375 used -> great portrait lens along with macro ability (1:1)

Tamron
90mm f2.8 macro - ~$350 used -> one of the best macro lenses - also great for portaits