Almost the instant I entered the classroom and sat down, the chime signaling the start of class began to ring out.
Literal
Classroom [into-に] enter-[and-て] seat [at-に] take [nominalizer-の] [and-と] almost same-time [at-に], start-of-class [obj-を] notify chime [gen-の] sound [subject-が] rings-out.
A tight temporal coincidence. 席に着く ('to take one's seat') is a fixed collocation — literally 'to arrive at the seat.' ~のとほぼ同時に is the formal temporal correlation 'almost simultaneously with X, Y' — combining の (nominalizer) with the particle と ('with') and 同時に ('simultaneously'). 始業 ('start of class / start of business') is the formal noun. 知らせる ('to notify, to inform') modifies チャイム as a relative clause. 鳴り響く ('to resound, to ring out') is a compound verb.