When your host can't reach an address that has to be routed but can reach the LAN that it shares a subnet with:
1) Ping localhost, same as pinging yourself by IP because localhost is resolved in /etc/hosts...somewhat arbitrary but it's nice to see something work, and to start at the bottom. I think it shows the IP stack is set up, it always is.
Your addressing is fine, if the host didn't have an IP address it wouldn't be able to reach anybody...unless it's using a different configured IP address in a subnet that the default gateway does have an IP address in to reach the other lan devices. But if you've only got one internal subnet then don't worry about that.
2) Ping another device on that same subnet, preferably the default gateway because ICMP probably won't be blocked by an internal IP/interface on a firewall by default. Lots of operating systems will drop ICMP and will make it look like you're not reaching them. So if you have to, check and disable firewalls, or at least allow ICMP. In any case, pinging the default gateway's IP on your 192.168.100.0/24 subnet shows the router's IP is set up, and since it doesn't require any routing, your host should be able to reach it without any routing set up at all on host or gateway, the default gateway setting on your host doesn't even have to be set up for this to work because the communication doesn't require routing.
3) Ping the external IP address on your default gateway. This is the one that connects to the internet. It shows that the default gateway configuration on your host is set up correctly.
4) From your gateway, if you have a router that can send pings, ping a DNS server, like 22.214.171.124. This shows that your router is connecting to the internet and has access to your hosts configured DNS server.
5) From Your host, ping 126.96.36.199. This shows that the default gateway is routing for your host, and that your host can reach DNS. Gateways tend to allow ICMP replies by default, but a firewall may drop ICMP echo requests even if they are initiated internally, or ICMP all echo replies.
6) ping www.google.com, or some other website that you like poking that doesn't drop ICMP. This shows that DNS resolution is working.
7) If all that works then you've likely got a firewall problem.