While designing IT Network Infrastructure, Which device shall we connect directly with the ISP? a Firewall i.e. Cisco ASA or a Cisco Router?

User Exec Mode (distinguished by the ">" prompt) is your first mode, which is used to get statistics from router, see which version IOS you're running, check memory resources and a few more things. 2-Privileged Mode (distingushed by the "#" prompt) is the second mode. Here you can enable or disable interfaces on the router, get more detailed information on the router, for example, view the running configuration of the router, copy the configuration, load a new configuration to the router, backup or delete the configuration, backup or delete the IOS and a lot more.

The ISP will give you a modem and if you are receiving via fiber optics will also give you a fiber "DIO" wich will send an ethernet connection to their modem/router. That will have an external IP address with you will configure as your WAN gateway on your firewall.

In the case of Cisco you ASA will be doing the firewall role.

Router's .. to route packets "in - out" and then filter it by firewall