Acts 11:26 records that believers were first called 'Christians' (Χριστιανοί) in Antioch. It appears at first to be a name given by outsiders to the group, and not a term that is used generally within the New Testament itself to refer to believers. In terms of derivation/composition, it is probably patterned on other terms used for political religious adherents of a particular persons. Much like we might come up with 'Clintonites', 'Reaganites', etc..

'Christianity' as an abstract is much later. Partly because it's a latinate term, and latin was not particularly inclined to construct these kinds of abstract terms for a religious system. It's derivation appears to be from Christus to Christianus (an adjective form of 'Christian' derived in the same was as the Greek term/in parallel to the Greek term). to Christianitas, a kind of 'abstract state of Christianness'.

I don't have a first appearance for the latter term though, and in any case it's first appearances seem to be linked to what we think of as 'Christendom', that is the universal social sphere of Christian society.

edit:

/u/CommonSensai is not wrong, though their comment lacks expansion. 'the way' does appear to be a term used by Christians to talk about their own belief system in its emergent context. This too is attested in Acts, for example, 19:23, "At that time a great disturbance took place concerning the Way." (NET), or Paul speaking in Acts 22:4.