UE sends measurement report(s) to eNodeB, based on the report(s) , eNodeB initiates either X2 or S1 based handover procedure. MobilityControlInfo within the RRC connection reconfiguration carries target physical cell Id.
Soon after this message, UE starts searching target cell based on PSS and SSS since both are used to identity physical cell Id. Once RRC connection reconfiguration complete message is received by eNodeB , handover procedure is considered successful. There are so many places like UE logs, target eNodeB logs from where can be figured out, UE is handed over to target cell successfully. Wireshark logs will also show change in "IP address" for GTP packets between eNodeB and SGW since eNodeB has been changed for an UE.
Now and days, few applications are there which can be used to get network information like current cell, tracking area code , plmn etc.