Titanium's status as an exotic metal might precede it, but the metal itself actually isn't terribly expensive. Part of the reason for the material's high historical cost is the fact that most of it comes from former Soviet-bloc countries, which made obtaining it difficult. However, manufacturing costs account for much of titanium's high market value, as titanium is very expensive to produce and difficult to machine.
Most metals come out of the earth as an oxide, or a form of "rust." Titanium oxidises or rusts at very high temperatures, so any machining has to be done in a pure noble gas atmosphere. Additionally, titanium has to be converted into four different compounds before it can solidify without oxidising. The end result is a material that far lighter than steel but is ultimately stronger and more elastic.
Titanium isn't the hardest material on Earth, it's just very tough for its weight. You can cut titanium using any kind of high-alloy, hardened tool steel like grades M1, M2, M7 and M10 or tougher grades like T5, T15, M33 and M40. Such tooling steels will work, but best results are generally obtained using purpose-made carbide steels like C-2 and C-3.
Cutting with Abrasives
Titanium is just as susceptible to abrasive forces as any other metal, and will cut using any abrasive that will work on steel. However, titanium grinds more like aluminium than steel, meaning that it's prone to clogging up abrasives like soft cheese rubbed on hot concrete. Both cutting titanium with a metal saw blade and grinding with an abrasive require a few special processes to keep the blades and abrasives from clogging up or wearing out.
Titanium doesn't conduct heat very well, so heat build-up when cutting tends to remain very localised. This heat build-up can cause the titanium to melt, clog and oxidise instead of cleanly cutting. The machine titanium, you'll have to use a very coarse abrasive or large cutting teeth to reduce clogging, and low cutting speeds and plenty of cutting fluid to keep the material cool. Your tools need to be very sharp and the cutting table or grinding set-up needs to be very rigid and stable to keep the titanium from deflecting during machining.
Cutting Off a Titanium Ring
As the USSR collapsed and titanium got cheaper, jewellers began using the material to make everything from wedding bands to earrings. Titanium's lack of reactivity and resistance to corrosion means that it will stay tarnish-free almost indefinitely, and it's a truly hypoallergenic material. At some point a rumour got started that titanium rings were impossible to cut off. This rumour likely came from ring retailers themselves, as gold and silver rings offered a far higher profit margin than the less-expensive titanium. While titanium is more difficult to cut than gold (which is very soft), it isn't much more so than silver or stainless steel. Any emergency room, ambulance or fire brigade and most jewellers will have the tools to quickly and painlessly cut any titanium ring from any finger.