Cutting Lapis lazuli causes a distinct unpleasant smell caused mainly by the sulphur content of the stone. Experienced cutters can tell the intenseness of the stones blue colour by the smell.

When polishing this stone, it must be handled with care on account of its modest hardness and not subject it to much pressure.
Another factor which make working with this stone more difficult is due to the fact that the quartz and pyrite inclusions are much harder then the rest of the stone.

Gemstone jewellery artisan Brent De Santis has the following suggestions for working with Lapis in his blog:

Lapis Lazuli inlay process for Jewellery

Lapis Lazuli is one of my favourite stones to work with because it is so beautiful while, at the same time, so challenging. What I have found with this stone is that the more Pyrite the Lapis contains, the more vulnerable it is to fractures. I generally cut a lapis stone into small slabs approximately 3mm in thickness and I look for fractures and cracks.
Once I notice traces of fractures or cracks I will cut through as many as I can with a diamond saw. I then let the smaller Lapis slabs sit in paraffin wax under a heat lamp with moderate heat to help give the stone a better polishing surface. To make sure I don't crack the stone while grinding, I use a 220 grit soft back diamond wheel to shape the Lapis, and if I want to inlay this stone into a straight sided channel, I will lightly hit the side of the stone on a 220 steel back wheel which will always give the stone a perfect straight edge. The advantage of a straight edge on any given stone is that it looks more professional especially when inlaying multiple stones next to each other.

The channels that I inlay are a little over 2mm in depth which gives me plenty of room to work with. The epoxy that I use is industrial strength gel type epoxy, where the consistency is thick enough to be easily removed with an exacto blade along the sides of the inlaid stones.
Once the stone is inlayed and the glue is cured, the grinding begins. I will grind away the stone starting with an aggressive wheel such as a 220 and step down to a 600 grit, then I start the polishing with the 1200 grit up to the 3000 grit wheel and then firmly hit the stone on a sapphire powder lap. After each step of grinding and polishing I always check for fissures or cracks. If any exist, you'll need to remove the stone and start the process from the beginning. I hope this can be of some help for you novice inlayers.

Lapis lazuli is somewhat porous and should be protected from chemicals and solvents, never put Lapis in a chemical dip or ionic jewellery cleaner and it should always be protected from acidic substances!.

Lapis lazuli is often sealed with colourless wax or synthetic resin. As long as these substances are not mixed with any colouring agent, this sealing process simply has the effect of improving the stone's wearing qualities.

However there is no need for the wearer to worry: a lapis lazuli that has grown matt from having been worn too much can easily be re-polished at any time.

Cleaning:
Your best bet is to simply wipe it off with a barely damp cloth without soap, micro fibre optical lens cleaning cloths seem to work best.
Lapis is not very hard at 5.5 and should be protected from other jewellery when stored to avoid scratches (i.e.. clean and store it as you would a pearl or Opal).