You should definitely contact HR. You should consider contacting your Insurance Provider help line as well. When I separated earlier this year I got different answers and it took a while to get everything straight. My circumstances were extremely unusual.
Internal HR website has the schedule of payments. You will pay the administrator, Workplace Resources, and the price is you existing premium, the Cisco subsidy and a five percent administrative fee. The administrator will mail out the forms but you can go to their website and fill out the form online w/o waiting for the paperwork to arrive. Either way your COBRA coverage backdates to the end of Cisco coverage (last day of your termination month). If anything happens during that period you can negotiate with your MD to delay submitting the paperwork or pay out of pocket and get reimbursed. You will likely get stuck with more out of pocket if you pay as the insurance company will negotiate a price and you may have to pay retail.
You can pay online and you can make multiple payments at one time. COBRA is good for 18 months (longer in some special circumstances) and it ends on the last day of the last month that you pay. If you get other coverage I think that your coverage under COBRA will become secondary. If this is likely to occur you should check with the Insurance company (Cisco Benefits doesn't know much about this).
The insurance is the same as you have now unless you change it during open enrollment which may be over. Same card, same group number, same everything. Does not reset your out-of-pocket clock. If you or covered family members are undergoing treatment for serious or chronic conditions this ensures continuity. hmm....it ensures continuity even if you never seek treatment :)
COBRA looks expensive until you price out similar coverage. Cisco has a very large risk pool and has done a good job negotiating price and benefits and pharma coverage with Cisco is particularly strong. Several folks in the SPVSS divestiture effectively took a $20K+ cut in pay because NuCorpse's insurance program did not cover meds they needed for chronic conditions.
Good luck.