What we usually call auto-negotiation is the process by which 2 devices agree on the link speed and duplex. This is really fast (almost instant from a human point of view) and, while issue with this process was not rare a decade ago, it is quite uncommon nowodays.
Your issue is most probably with spanning-tree, which is enabled by default, in PVST+ mode, on those switches, with a delay of 15 seconds before a port goes to the forwarding state.
You should be able to see the status of the port with the command
show spanning-tree interface ethernet <your interface number>
(the exact syntax may vary, I'm not this familiar with those switches)
If this delay cause you some trouble you can configure the ports that are connected to end devices with the command
spanning-tree portfast on the chose interfaces. This will instruct the switch to immediately transition the port to the forwarding state when the link connect.