P.S. never heard of titanium chips catching fire. Now magnesium is a different story...
It usually takes a bit of effort, but you can light up the titanium chips on your machine. At work, it happens occasionally.
As metioned, a Class D extinguisher is recommended (Ti will burn in air, water, nitrogen, carbon dioxide, silica). In the event of a titanium fire, we use salt. Lots of salt. When you think the fire is extinguished, LEAVE IT ALONE and let it continue to cool.
All of the tooling types mentioned here will do just fine, but if you can give us a better description of some of the machining operations and dimensions you'll be making, we may be able to recommend better tooling that is better for cutting or better on your bank account. We mostly use carbides, both brazed and coated inserts. We machine it dry or with flood coolant, it depends on the machining operation, the type of work piece, the alloy, and the condition (cast vs. forged vs. rolled).
What alloy of Ti will you be working with? CP? Ti 6-4?
Most alloys are gummy, but machine OK with the right set up. Low speeds with adequate feeds and a machine that has enough low end torque to keep the tooling moving help. Titanium is extremely elastic and has a memory. The memory effect makes the material want to go back to where it came from at the last hot working. For example, when saw cutting a piece of billet, sometimes it springs closed on the saw blade, sometimes it opens up. Smaller diameter round bar will go out of straight when turned due to the hot rolling process to produce the bar. After mechanical straightening, it will stay straight.....for a while, then go back to where it came from.