A few other things worth trying is to set your inside router into bridge mode, which will cause it to function as a repeater rather than a router, which would save you from having to deal with 2 separate internal subnets. But if you want to allow connections to one box from the internet, this may be a bad idea as you will no longer have the extra layer of security protecting the other machines on 192.168.1.0/24.
You don't use the /24 notation when you set a default gateway, only when setting a netmask.
Basically, what the /24 at the end is telling you is where to draw the line between the network address and the host address. This all goes way back to methods that are no longer used much, but long long ago, you could get 3 classes of addresses, and what differentiated them was how many hosts you could put on a single subnet. A class A network used the first octet for the network part of the address, and the next 3 for host address. so N=network, H=host would look like N.H.H.H for class A. This allowed over 16 million hosts per subnet. Not very practical. A class B network would split the network and host halfway. N.N.H.H, the corresponding netmask would be 255.255.0.0 or FF.FF.0.0 or /16, those are just 3 ways of saying the same thing, the first one uses decimal, the second uses hex, and the third /16 is counting the number of binary ones. If you are totally confused by now, which you should be, because these different ways of saying the same thing are used pretty interchangeably and probably seems pretty random. Especially if you aren't fluent in base2 or base16 number systems. so to recap, decimal 255 is the same as saying hexadecimal FF, which can easily be converted to binary, or 11111111, which is 8 ones and may also be written down as /8 (the last /8 notation is only used for masking bits, not for much else).
If you want to reach a server on your own network from the outside, you must use a NAT, or network address translator if your ISP only assigns you a single IP address (which is taken by your modem/router). the NAT will keep track of which internal addresses are associated with certain connections going out to the internet. If you have more than one machine internally which connects to the internet, then you already have a functioning NAT in place.
What you need to do next is set up a port forward on your router/modem that will point to the IP address of your server, and the desired port. If you want to run a web server, you probably want to map all incoming connections from the internet to your modem on port 80, and forward those to your server's port 80. If you want to set up a different service, just change port 80 to appropriate port number. You may need to do this for both TCP and UDP connections. It should be fairly easy to figure out if you find the right menu on the modem/router. if all else fails, RTFM.