Nmap

Scan ports on IP

nmap -sT <IP>

Scan ports on subnet

nmap -sT <Net/cidr>

Scan for vulnerabilities

nmap --script vuln <IP>

Scan for specific port

nmap -sT -p <PORT> <IP>

Remarks

If you want to check your public IP, you need to run nmap from a location outside of your LAN.