Titanium can be MIG or TIG welded. Any filler metal or wire should be the same material grade as what is being welded. You can subsitute lesser quality in a pinch, but it's not recommended.
The joints have to be super clean---clean beyond any realm of cleanliness ever before seen in welding. Very clean. :-laf
Nitrogen, oxygen, and hydrogen cannot under any circumstance be introduced into the arc or puddle--titanium reacts with these gases in it's molten state and becomes very brittle. 99. 99% pure argon shielding gas with a dew point -50*C (very, very dry) or lower is pretty much a requirement.
Anytime you stop the arc you must continue to shoot shielding gas on the bead/puddle until it cools from a red hot/molten state, and trim the oxidized end of the filler metal or wire before starting again--otherwise you'll ruin the bead.
In a nutshell... .....